The full story
In August 1945, Logan is a POW in a Japanese camp outside Nagasaki when the atomic bomb hits the city. He shields a young officer, Ichirō Yashida, from the blast.
Present day—two years after being forced to kill Jean Grey to save the world in X-Men: The Last Stand—Logan is living as a hermit in the Yukon, tormented by hallucinations of her. Yukio, a mutant who can foresee a person's death, finds him and brings him to Tokyo on behalf of the dying Yashida, now the head of a technology empire, who wants to repay his life debt before he dies.
Yashida offers to transfer Wolverine's healing factor into his own body—ending Logan's near-immortality and letting Yashida live. Logan refuses. That night, Yashida's physician Dr. Green secretly poisons him.
Yashida dies the next morning, naming his granddaughter Mariko—not his son Shingen—as heir to the company. At the funeral, the Yakuza attempt to kidnap Mariko; Logan gets her out, but his wounds are healing slower than they should. They fight off more Yakuza on a bullet train, hide out in a love hotel, then travel to Yashida's house in Nagasaki, where Logan and Mariko fall in love—and where Logan learns Mariko's fiancé, a corrupt government minister, conspired with Shingen to have her taken.
Ninjas led by Harada attack the house and capture Mariko. Logan and Yukio discover a robotic device attached to his heart, suppressing his healing factor; Logan tears it out himself. Shingen attacks during the extraction, and Logan kills him.
Logan tracks Mariko to Yashida's ancestral village, where Harada's ninjas capture him and Dr. Green straps him into a machine built to drain what's left of his healing ability—introducing the Silver Samurai, an armored, katana-wielding suit built from adamantium. Mariko frees Logan from the machine; Harada, realizing he's been serving the wrong side, dies helping him escape.
Yukio kills Dr. Green. In the final fight, the Silver Samurai severs Logan's adamantium claws and starts pulling his healing factor directly—and is revealed to be Yashida himself, who faked his death to build the suit. As the drain restores Yashida's youth, Mariko stabs him with Logan's own severed claws. Logan, healing again, regenerates bone claws and throws Yashida from the platform, killing him.
Logan has a final hallucination of Jean and lets her go. Mariko becomes CEO of Yashida Industries; Yukio stays on as his bodyguard.
Mid-credits, two years later: Logan is approached at a US airport by Erik Lehnsherr, who warns him of a weapon humans are building that would end the mutant race. Charles Xavier—presumed dead since The Last Stand—is with him, alive. The weapon is the Sentinels, and the warning is where X-Men: Days of Future Past picks up.