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The Last Pagan in London
The earliest known ancestor of Wolverine was Folkbern Logan, who lived in the early 11th century A.D. Around the year 1013, Folkbern was an English night watchman stationed on London Bridge and was considered to be a somewhat of an oddity for his time, as he still worshipped the old gods. This unusual trait gained him the title “The last pagan in London.” One night, while he was manning his post and discussing all the ale and wenches he planned to indulge on once his shift was finished, he was attacked by that time period’s Horsemen of Apocalypse. Apocalypse had dispatched his Horsemen to kill Folkbern, as he had been told that a descendent of his would become one of his greatest enemies. As Folkbern was about to be slain, he prayed for the old gods to deliver him a savior. Just as Folkbern finished speaking, Thor, the Norse god of thunder, appeared on the scene and ended the threat of the Horsemen and Apocalypse, ensuring Folkbern's survival. [Uncanny Avengers #6] Folkbern's descendents eventually made their way to Canada.
The Big House on the Hill
By the 19th century, Folkbern's line had borne Thomas Logan, who lived in Alberta, serving as grounds keeper on the infamous Hudson Estate. Before he had died at the age of 18, Thomas' father had produced two sons other than Thomas, though they did not seem to be a part of Thomas' life. Whether he shared the same mother with his other brothers, Thomas' mother was a crass woman whose "offensive and unnerving acts" possibly led to her being stabbed to death in the middle of town in broad daylight. Thomas himself was a very unhappy and cruel man who repeatedly abused alcohol and beat his son, whom he referred to as Dog.” It's possible Thomas' animosity arose from being abandoned by Dog's mother, who departed for parts unknown immediately after her son's birth.
Thomas also bore a particular resentment for his employer, John Howlett, despite his kindness, likely a result of his desire for his master's wife, Elizabeth. Elizabeth Hudson had been fortunate and married into a family of great wealth with the Howletts. She and her husband, John, had one child prior to the completion of the Hudson Estate, John Howlett, Jr. The boy had developed claws prior to his 13th birthday and either died or was murdered as a result. At some point, Elizabeth began an affair with Thomas Logan and he fathered her second son, James Howlett, later known as Wolverine. John was apparently oblivious to the true paternity of his son, and neither Thomas not Elizabeth gave him reason to believe otherwise. While Thomas was the father of James, it seems unlikely that he was the father of John, Jr. As John had been born prior to the completion of the Hudson Estate, a groundskeeper would not have been needed before that time and it seems improbable that Elizabeth, a woman of means, would have met Thomas beforehand. Regardless of John's true father, his death sent Elizabeth into a deep depression and her husband had her committed for a time. When Elizabeth returned, she became a recluse and neglected her son. In the hopes of giving his son some companionship, John hired a girl named Rose from the nearby town to serve as his governess.
Rose became fast friends with James and Dog. The trio became inseparable for a time, both boys unaware they were in fact half-brothers. As all three grew, tensions began to mount. The abuse that Dog had suffered as a child warped his mind and he murdered James' puppy in front of him. James informed his father John of the incident and he had Thomas and Dog removed from his estate. Thomas was in a rage at this turn of events. He armed himself and his son and attacked the Hudson Estate to finally claim John's wealth and his wife, Elizabeth. During the ensuing melee, John was killed. When Thomas made a move on James, the stress prompted his mutant powers to develop and claws burst through his knuckles, impaling Thomas and killing him. Elizabeth was so horrified when she saw James sprout claws that she took her own life. A very confused James ran out into the wilderness, pursued by Rose. While both were away from the estate, Dog named Rose the culprit behind the murders and a warrant was issued for her arrest. With no choice, Rose and James, the latter now under the synonym “Logan,” fled to British Columbia.
Logan and Rose managed to build a life up in the wilderness. Rose found love with a man named Smithy, who also took on a fatherly role to the recently orphaned Logan. This peace 
would not last, however, when a grown up Dog tracked Logan and Rose down. During a fight between Dog and Logan, Rose was inadvertently killed with Logan's claws. This event caused Logan to lose grip on his humanity and he fled further into the wilderness, where he would remain for quite some time. [Origin #1-6] When Logan eventually returned to humanity, he had repressed all knowledge of his earlier life.
Dog, on the other hand, continued to hunt Wolverine but now he was hunting him throughout time and space. He had stumbled upon “time diamonds” in a mine near the Yukon River. [Astonishing Spider-Man & Wolverine #3] Since he's journeyed to the future, he's spared with his half brother many times. One such time he hunted Wolverine and his students throughout the Savage Land. This brought him to the attention of the villain Mystique, another person who despised Wolverine, and she recruited him to be a teacher at her Hellfire Academy. [Wolverine and the X-Men #25-28, 31]
Frederick's Line
Wolverine's mother had two brothers, Elias and Frederick Hudson. Tragedy would follow them just as it had done their sister. The two men were the last of the Hudson line after their sister's suicide and the untimely death of their parents, who had drowned when a ship they were on sank off the coast of Vancouver Island. Elias was a wealthy man who owned the Hudson Bay Company. Wolverine briefly worked for his uncle, both seemingly unaware they were related. A short time after Wolverine left the company, Elias took his own life.
Elias’ brother, Frederick, worked for a training facility that employed Silas Burr and his own nephew, Wolverine. Though Frederick never encountered Wolverine, he did cross paths with his son (and Frederick’s great-nephew), Daken. Unfortunately, for Frederick, he did not survive the experience.. Neither Elias nor Frederick sired a legitimate heir but Frederick had fathered an illegitimate son by his secretary, Caitlyn MacDonald, prior to his demise. In an act of spite, Caitlyn put Frederick's name on the birth certificate and named her son Frederick Hudson the 2nd. Two days after this, Caitlyn was found hanging from her hospital room window, apparently having committed suicide.
The boy was raised in an orphanage. By all accounts, he was volatile and was dead by the age of 30, killed in a bar fight. Before he had died, however, he managed to sire three sons by three women, though it's not clear if any of these men knew of each other's existence. The eldest was Truett Hudson. Truett went on to become the Professor of the Weapon X program responsible for subjecting his cousin Logan to the adamantium bonding process. Truett would be murdered a significant time later by another one of his former test subjects, Silver Fox. [Wolverine (2nd series) #50] The second son named Victor Hudson was the product of Frederick's relationship with a young religious woman named Bernice. The poor lad had been unfortunate at birth and was afflicted with blindness and mutism. A short time after his birth, his mother walked into a church, claiming her son was demon spawn. Just before she shot herself in the head, she claimed her son had been taken by “the devil himself.” Bernice's grandiose description notwithstanding, the child had indeed taken by a devil of sorts, the ancient villain Romulus. Under Romulus' guidance, Victor became a enforcer. Frederick's youngest son, James, however, took a more heroic path than his siblings. [Wolverine Origins #33]
Canada's Mightiest Superheroes
James Hudson had also been born with a keen scientific mind like his half-brother, Truett. Unlike his half brother, however, James decided to use his intellect to help his fellow man. James had been influenced by the American superhero team the Fantastic Four and thought that Canada would need its own superhero team to protect its borders. To that end, he created Alpha Flight, a diverse team of Canadian mutants, mystics, gods etc. James served on the team as their leader under the guise of “Guardian,” using a cybernetic suit of his own design that granted him flight and enhanced strength, amongst other abilities. His relative and distant cousin, Wolverine, also briefly served with the group, neither aware of their relation at the time. [Alpha Flight (1st series) #3, 8, 11, 17]
Unfortunately for James, at one point Alpha Flight's government sponsor Department H was taken over by a corrupt council, which viewed him as a loose cannon. To that end, they attempted to kill James. However, they felt that Canada still needed a “public” superhero, so they copied James' memories into an adolescent clone of himself and jettisoned the “real” James off into space. Department H managed to pass off the young James as the real one by pretending James’ new youth was a result of a lab accident. [Alpha Flight (2nd series) #19-20] When the original James made his way back to Earth, however, the truth came out. Despite some initial antipathy between the two, both men managed to work alongside each other, with the clone becoming Vindicator to James' Guardian. The young clone later perished in a battle against M.O.D.O.K. and A.I.M and was mourned by his progenitor, James. [Wolverine (2nd series) #142-143]
In addition to his brothers and his clone, James has other family, beginning with the woman whom he married, Heather MacNeil. [Alpha Flight: In The Beginning Minus #1] Heather Hudson has also served in Alpha Flight using various suits of James' design, first as Guardian during a period that her husband was believed dead and later Vindicator when he returned. Over the years, the marriage has been strained. James' multiple “deaths” took their toll on the relationship and at various points Heather entered into relationships with other men, which caused issues when James reappeared. Despite all of their marital problems, James and Heather did have some good times together, most notably the birth of their daughter, Claire. [X-Men Unlimited (1st series) #45] When both James and Heather died in a battle with the Collective, custody of Claire went to Heather's cousin and his wife, David and Yvette MacNeil. [New Avengers (1st series) #16] However, in the aftermath of a battle between gods known as the Chaos War which briefly weakened the veil between life and death, James and Heather were both resurrected. [Chaos War #5] James and Heather's first action was to attempt to regain custody of Claire. The courts ruled in favor of the MacNeils, viewing the Hudsons’ status as superheroes too dangerous to their daughter's wellbeing. Both parents were devastated, leading Heather to later fall under the influence of the Master of the World and his “Unity” political party. While under his control, Heather murdered her cousin and his wife and abducted Claire. James and the rest of Alpha Flight managed to help Heather break free of the Master's influence and together they defeated the Master and his Unity group. After the team’s victory over the Master of the World, it appears James and Heather have finally separated. James has remained with Alpha Flight while Heather has seemingly taken the words of the court to heart and retired from her role as a superhero to raise her daughter. Single-handedly. [Alpha Flight (4th series) 1-8] 
Daken, the Bastard Child
Despite the seeming curse on the Hudson bloodline, Logan has had periods of happiness. Shortly after World War II, he met a Japanese woman named Itsu, from a place called Jasmine Falls. The two had a whirlwind romance and married. This happy time unfortunately came to an end shortly after Itsu found herself with child. Having long been interested in Logan, the shadow power broker Romulus arranged for Itsu’s murer, after which Romulus ripped the baby from her womb. When Logan found his wife's dead body, he believed that their child must have died with her. [Wolverine (3rd series) #38, 40] However, as a result of his inherited healing factor, the baby did survive and was placed on the doorstep of a childless Japanese couple named Akhira and Natsumi. The couple took this as a good omen and named the boy, Akihiro.
Akihiro was a difficult child who battled with his psychopathic urges very early on. The fact that local children dubbed him “Daken,” a Japanese slur meaning Mongrel, only added to his anti-social behavior. Akhira's mother Natsumi, unlike her husband, could recognize the darkness in her son and it terrified her. Her feelings for him further soured when she found out she was pregnant. Natsumi went on to give birth to a boy named Junchiro. Akihiro was filled with resentment and murdered his younger brother when he was a few months old. When his father found out what had happened he renounced his surviving son and told him he could no longer have the name Akihiro. Far from remorseful Akihiro claimed it was never his real name anyway. He was always Daken. During this argument an enraged Natsumi burst out of their family home brandishing a weapon, lunging toward her son. Just as his biological father before him, Daken claws exploded out of his knuckles, killing his attacker. Akihira gripped the body of his dead wife and told his son that he was weak and took his own life. Just as Daken looked upon the bodies of his adoptive parents, the ancient mutant Romulus revealed himself from the shadows. [Wolverine Origins #26]
Under Romulus' tutelage, Daken's mind continued to warp. Romulus indoctrinated the boy with a great hatred of his biological father, Logan, by blaming his weakness as the cause of the death of Daken's mother, Itsu. Years later, acting independent of Romulus, Daken would go to great lengths to torment his father, even after learning the truth of his mother’s death. Throughout Daken’s betrayal of him, Logan hoped for a future reconciliation with his son. However, learning from a trip to the future the horror that would happen if he did not stop his son, Wolverine saw no alternative but to kill his son. During a fight between Wolverine’s X-Force team and Daken’s own Brotherhood of Evil Mutants, Wolverine took the opportunity and drowned Daken in a shallow pool of water. [Uncanny X-Force (1st series) #26-34] Daken's corpse was later recovered by Uriel and Eimin, the children of Archangel, who resurrected him using the life seed and made him one of their Horsemen of Death. [Uncanny Avengers #8]
The Gaijin and the noblewoman
Decades after the death of Itsu, Wolverine would find love with another Japanese woman, Mariko Yashida. While visiting his former teammate, Sunfire, in Japan, 
Wolverine met the atomic samurai's cousin, the noblewoman Mariko Yashida. [X-Men (1st series) #118-120] It was love at first sight for the foreigner but the course of true love did not run smoothly. Shingen, Mariko's father, objected to the relationship and Wolverine was forced to kill him in self defense. With Shingen dead, Mariko became the head of her clan and she and Wolverine became engaged. Later, the villain Mastermind started to manipulate Mariko's feelings, forcing her to call off the wedding and align her clan with the Yakuza and her criminal half-brother, the Silver Samarai. When the manipulations were revealed, Mariko no longer considered herself worthy of Wolverine and vowed that she would not marry him till she had redeemed the honor of herself and severed her clan's criminal connections. [Wolverine (1st series) #1-4, Uncanny X-Men (1st series) #174-176]
Sometime later, Wolverine and the X-Men were again in Tokyo, Japan fighting a dragon when he came across a mother and daughter trapped under some rubble. As the mother lay dying, she asked who would take care of her daughter, Amiko. In that moment, Wolverine made the woman a solemn vow that he'd raise the girl as his own. 
Despite their separation, Wolverine ultimately entrusted Amiko's care to Mariko and would regularly visit the two. [Uncanny X-Men (1st series) #181]
Sometime later, in an effort to cut all ties her family had with the underworld so she could marry her beloved, Mariko attempted to sell all of Clan Yashida's illegal holdings to crime lord Matsu'o Tsurayaba. However, when Wolverine coincidently arrived in Japan, Matsu'o viewed this as an act of bad faith. As a result of this, he hired a poisoner named Reiko to assassinate Mariko. [Wolverine (2nd series) #55-57] With Mariko dead, Amiko went into the Japanese foster care system for some time, until Logan and Harada secured her placement with Yukio instead of Wolverine. [Wolverine (2nd series) #82]
Yukio and Amiko's bond seems stronger than the one she shared with Mariko or Wolverine. When Amiko and Yukio were attacked by a demoniacally possessed Wolverine, both women referred to each other as mother and daughter. Yukio suffered a spinal injury after Wolverine's attack and afterward lost the use of her legs. Despite the disability, the duo have become somewhat of a crime fighting duo in Japan, occasionally teaming up with Wolverine. [Wolverine (4th series) #2-3 and Wolverine (2nd series) #300-304]
Spreading his seed
Over a century old, Wolverine has had known many women and has, on several occasions, left more than memories behind with them. In addition to Wolverine's legitimate children, Daken and Amiko there are six known “bastards” to the X-Men's Canucklehead.
One such child might be the product of the union of Wolverine and Gahck, a former chieftain of one of the many Savage Lands tribes, the Tribe of Fire. 
During a trip to the Savage Land, Wolverine's plane took damage and he crashed. With his healing factor, he easily survived but, while he was unconscious, the Tribe of Fire claimed the wreckage. In an attempt to reclaim the plane so he could return home, Wolverine was forced to defeat Gahck in combat for leadership of the tribe. To Gahck's credit, she put up a good fight but lost due to Wolverine's superior skills. That night, Gahck went to where Wolverine was resting to try and ascertain whether he was, as her people's believed, a god. Despite realizing he was a mere man, she decided to sleep with him anyway. A short time later, Wolverine encountered and defeated a robot duplicate of Apocalypse. Using a plane that was within the robot Apocalypse's citadel, Wolverine fled the Savage Land abandoning Gahck and his new people. Sometime later, Gahck was seen with a young child of indeterminate gender. It’s unclear whether this child was actually hers, or whether it is a child she shared with Wolverine. [Wolverine: The Jungle Adventure]
The Mongrels
The Mongrels were a group of mercenaries and rogues brought together by Daken to aid the quest of revenge, sought by the group known as the Red Right Hand. Although none of them seemed to be aware of their shared relation, each were forgotten bastard children of Logan’s. The Red Right Hand was made up of family members and friends of the people Wolverine had killed over the years. Seeking revenge, they were offered an opportunity by Daken, who sought to trick Wolverine into killing his own children, so he would experience loss as they had. So driven were they in their quest that the RRH planned on preventing Wolverine from getting any retribution by taking their own lives once Wolverine had killed the Mongrels. Very little is known about the children they recruited, save that there were five: two female (Fire Knives, Shadowstalker) and three male (Cannonfoot, Sawfist, Gunhawk), each of drastically varying ages. Based on comments from the leadership of the Red Right Hand, it appears not all of them were aware of Wolverine's identity as their sire. The Red Right Hand's plan succeeded without fail and Wolverine did indeed kill his five children in combat. When the Red Right Hand posthumously revealed the Mongrels' identity via a video, Wolverine was disgusted. It's unclear what he did with the bodies of the others but he did at least bury the eldest, William Downing, aka Gunhawk next to his mother in her home town in Canada. Though none of the children were apparently mutants, the whole affair occurred post M-Day, so they may have been depowered. [Wolverine (4th series) #1-4, 10-15]
Child or Weapon?
In addition to Wolverine's many, many children, there is also X-23. Technically, X-23 is not a child of Wolverine, though he regularly treats her as such. In actuality, she is a clone of Wolverine who was created by Doctor Martin Sutter, Doctor Zander Rice and Doctor Sarah Kinney.
Sutter had been a doctor at the original Weapon X facility that created the Adamantium boding process, along with Wolverine's cousin, Truett Hudson. 
Sutter was one of the few people to make it out of the facility alive when Wolverine escaped. One of the many slain scientists was Dr. Dale Rice, who was survived by his son, Zander Rice. With Dale dead, Sutter raised Zander himself and the youth followed in his father's footsteps and became a scientist of some note. He and Sutter decided they wanted to recreate the Weapon X project and created an organization called the Facility. The endeavor was met with problems, as only someone with a healing factor could survive the Adamantium bonding process. Sutter had managed to retrieve some DNA samples when he fled during Wolverine's escape and decided a clone was the way to go. To that end, they recruited geneticist Dr. Sarah Kinney.
Despite her previous moral ambivalence to creating a clone for experimentation, Kinney felt very maternal towards the child, dubbed X-23, once pregnant. This affection only grew when X-23 was born. On numerous occasions she would subvert Sutter and Rice's authority by reading her child fairy tales instead of the prescribed reading material, the Art of War. Kinney's presence notwithstanding, life at the Facility was a cruel one for X-23. Rice and Sutter came up with several inventive ways to torture X-23 and keep her compliant. One particular method they used to ensure that she obeyed was to create a trigger scent. The scent would prompt X-23 to fly into a murderous rage, ensuring her target died regardless of any moral objection she might have. By the time X-23 was in teens, Kinney had enough and she wanted to free her child of her servitude so she could raise her herself. She went to X-23 and explained to her that they were going to escape and she'd need to fight her way out and that she'd wait for her outside the Facility's gates. Despite Kinney's bravery against incredible odds, her plan ultimately failed. Rice had predicted Kinney's defection and exposed her to X-23 trigger scent. When X-23 did finally reunite with her mother outside the Facility’s gates after battling through a score of security staff, she flew into a psychopathic rage and fatally wounded Kinney. As wind dissipated the scent and she saw Kinney dying before her, X-23 was grief-stricken. With her last breath, Kinney told X-23 to remember that she wasn't a weapon but a person, that she was her daughter and her real name was Laura. [X-23 (1st series) 1-6]
X-23 took her mother's name Laura Kinney and fled the scene. The following years were very dark for Laura and she started to view herself and her sire, Wolverine, as monsters. Sometime later, she tracked him down in Westchester with the aim of killing the both of them, thinking the world would be better without them. Kinney had written a letter to Wolverine just before she died, explaining who X-23 was and with that knowledge he managed to calm Laura down. This was the start of some healing for Laura, as the two started to bond. Over time, Wolverine has become somewhat of a father figure to her. [X-23: Target X #6] 
Since then, Laura has followed in Wolverine's footsteps and become a notable heroine, serving alongside the X-Men and the Avengers at various points. On one such occasion, she teamed up with an eclectic group of heroes including the Red Hulk, Ghost Rider and Venom to fight the demon Blackheart when he attempted to bring hell to Earth. Ghost Rider managed to contain the contagion to Las Vegas and the four heroes battled Blackheart's forces to return Las Vegas to its original state. During this time, X-23 encountered four clones of herself that Blackheart had created and bonded with Venom symbiotes. As X-23 was technically in “hell” when she faced her clones, she easily killed all four as their healing factors didn't work in that realm. [Venom (2nd Series) 13, 13.2]