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Haven't really read the comix, but I'm somewhat familiar with the characters and everything. The story is about prejudice. Human beings fear mutants, who each have their own unique powers. There's a world summit coming up to discuss the issue. Meanwhile, in the U.S., a senator named Robert Kelly is pushing for a law requiring mutants to reveal themselves for a registry. Which of course would open them up to discrimination and hate crimes. Still, there are valid points, I suppose, on both sides of the argument. There is reason to fear some mutants: namely, the Brotherhood of Mutants, led by Erik Lehnsherr (aka Magneto, who uses his magnetic power to control metal). His followers include Sabretooth (who I don't know how to describe except to say he's fairly strong, tough, and feral), Mystique (who can change her appearance to look like pretty much anyone), and Toad (who leaps around and has a long, sticky tongue). Magneto believes there is a war coming between mutants and the rest of humanity, and he intends to fight and win that war.
Meanwhile, his old friend Prof. Charles Xavier (who can read and control people's minds; he also uses a machine called Cerebro to enhance his powers, in order to telepathically search for new mutants to recruit), runs a school for mutants, and opposes him. Xavier sees hope that humanity will come to accept mutants. Some of his former students, including Ororo Munroe (aka Storm, who can control the weather), Dr. Jean Grey (who has telekenesis and some telepathic power), and Scott Summers (aka Cyclops, whose eyes emit a destructive ray, and he can only open them while wearing a special visor or glasses), are now part of a group of heroes called the X-Men. Soon two more mutants, Logan (aka Wolverine, who has great regenerative powers) and Marie D'Ancanto (aka Rogue, who absorbs people's life force when they touch her skin, as well as the powers of any mutants who touch her), are recruited. Wolverine can't remember anything about his life prior to 15 years ago, at which point someone conducted experiments upon him, grafting a super-metal called adamantium, onto his entire skeletal structure, adding claws he can extend and retract at will. Xavier promises to help him find answers about his past if he joins the group.
Magneto kidnaps Sen. Kelly, uses a machine to turn him into a mutant, then Kelly escapes, and is helped by the X-Men, but they discover the transformation is killing him, which Magneto doesn't know. Magneto then kidnaps Rogue, who he intends to use to power his machine to turn everyone at the upcoming summit into mutants. The X-Men have to rescue Rogue and stop Magneto's plan before he kills everyone in the area. Of course, they do. Magneto is locked away in a special plastic prison. Mystique takes the place of Senator Kelly, who the public doesn't know is dead. (S)he votes against the proposed mutant registry.
Xavier finds out about an abandoned facility at Alkali Lake, in Canada, which may hold answers for Wolverine. That's pretty much the story so far. And while the battles are cool and all, the most interesting thing remains the moral questions, as well as the personal struggles.