Since "X2: X-Men United" in 2003 when Jean Grey disappeared, a few other X-Men movies were made with Wolverine gaining more onscreen presence. "The Wolverine" made a decade later, is sort of a spin-off movie due to Hugh Jackman's excellent portrayal of the character.
After what appeared to be Jean's death, Wolverine went into hiding out of guilt. His recurring nightmares of Jean beside him, night after night, turned him into a recluse. That was when he was brought to Japan and met Mariko, his love-interest in the comics. I didn't quite like the storyline though, with the Yakuzas and ninjas and the Viper.
For me, the best part of the movie was the easter egg segment, where Magneto and Prof Xavier appeared, and hinted of a huge storm brewing ahead of them, thus uniting both forces together to fight what was coming up.
