For a man who sees himself as such a loner, Wolverine has gathered quite a family around him since his creation, in both literal and metaphorical terms (The price of being such a successful franchise, I suppose). Meet his kids - well, the ones we know about, at least.
In what still counts as one of the most "WTF" moments of Marvel's history, Wolverine's teenage female clone was introduced to comic audiences as a prostitute servicing those with particularly masochistic tendencies. Just think about that for a second. Luckily, saner heads prevailed, and pretty soon Laura Kinney (AKA X-23) left behind that world to become one of the X-Men, and currently appears in the regular X-Force series as a troubled goth kid, which befits her status as a faulty clone created by scientists trying to duplicate Wolverine but without that pesky conscience.
Much more straight-forwardly, Daken (AKA Dark Wolverine, although he doesn't actually call himself that, thankfully) is Wolverine's son. Much less straight-forwardly, he was stolen from his dead mother's womb, had his existence kept secret from his father for years and grew into the kind of man who'll kill for fun and probably listen to My Chemical Romance while cutting himself with his claws and feeling like the world doesn't love him. Call it Marvel's underhanded way of demonstrating its support for traditional family values. He's currently pretending to be his father as part of Norman Osborn's "Dark Avengers" team of n'er-do-wells, alongside Venom and other villainous types.
Many comic fans' first crush, Pryde was the spunky girl geek with superpowers that brought about one of the stranger running themes of Wolverine's career - his surrogate fatherhood. Acting against type, the grim loner Wolverine took the young Pryde under his wing when she first joined the X-Men, leading to comedic adventures and misunderstandings and only a few creepy scenes when you stopped to wonder why Wolverine wanted to hang out with a teenage girl. Thanks to Joss Whedon, Kitty's currently flying inside a giant bullet in space.
Surrogate daughter #2, Amiko was adopted by Wolverine after her mother died during a battle between the X-Men and a giant dragon. He passed her to various, more responsible, caretakers during the years, but has always remained in danger from Wolverine's enemies. Luckily, as incredible coincidence or lazy plotting would have it, she turned out to be descended from ancient Japanese warriors and is now studying with them to become an unstoppable fighting machine.
Surrogate daughter #3, and a reliable indicator of your age. Simply put, if you find Jubilation Lee (Yes, that's her real name) anything other than an irritating failed attempt to recreate the original relationship between Kitty Pryde and Wolverine but with a rip-off of Frank Miller's Carrie Kelly from The Dark Knight Returns, then you were either a teenager in the 1990s or her creator Chris Claremont. One of the first signs that Claremont was beginning to lose touch with his audience, Jubilee nonetheless was successful enough to make it into the 1990s X-Men cartoon, as well as numerous comic spin-offs from X-Men, including Generation X and the ongoing Wolverine series. These days, she's in limbo following the cancellation of her last series, New Warriors.
Scotty and Jade
In the alternate future "Old Man Logan" storyline, Wolverine has two kids, both of whom are rarely seen, but appear to be non-superpowered and relatively normal. Scotty is clearly named for Scott Summers, AKA Cyclops from the X-Men, but I admit that Jade is a bit more obscure. If any of you have any suggestions, don't be shy.
Again from an alternate future, but Fantastic Force's Hooded Man - a future Wolverine who's traveled back to our present - is the stepfather of Banner Jr., an intelligent descendant of the Hulk. Almost everything else about the relatively new characters is unknown.