Wolverine is a distant man with a past shrouded in mystery. With his memories reduced to a flashing series of contradictory images, Logan came to the Xavier Institute at the behest of Professor X. The object of fear, awe and speculation on the part of the students, he inspired the X-Men in good ways by providing battle tips after clandestinely watching Danger Room sessions and in bad ways in inspiring conflict between the core group of X-Men simply by virtue of his loner status and aggressive behavior. But, his softer side often showed through. He was sometimes a protective father-figure to Kitty Pryde (whom he called "half-pint"), X23, Kurt Wagner (whom he called "elf"), and Rogue. The closest of the four was Kitty, who believed that Logan wouldn't hurt her, even in a mind-control-like state that was supposed to cause him to kill her. When he thought she might have died, when she had actually just phased away, he was overcome with grief. He once stated to a mind controlled Kitty that he, in turn, could never hurt her. He also shared a close relation with X23. He was protective of her and thought mostly of her safety. He seemed proud of her abilities and deeply cared for her in a father-daughter way.
This Wolverine does not have any romantic interest in Jean Grey and does not have a love rivalry with Cyclops, due to the fact he's obviously older than the two. However in the X-Men Evolution comic book he does (only a little). In fact Rogue and Jean's feelings for Cyclops in Evolution closely matches Cyclops and Wolverine's feelings for Jean in the comics as Rogue expresses jealousy and resentment towards Jean for loving Scott like how Wolverine expresses jealousy towards Cyclops for loving Jean.
It is unknown if Logan has passed on or retired, since he is not seen in future flashes of Professor Xavier.
In X-Men Evolution, Logan takes more of a mentor role to Scott and the rest of the X-Men, he also briefly took command of the team on at least three or more occasions during the series.