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I'm not one to shoe-horn in terms of genre except where blatantly obvious, but Batman Arkham Asylum made me think rather quickly of classic zany 3D platformers such as Mario 64 and Banjo-Kazooie if they had been crossed with the gritty semi-realism of the Splinter Cell and/or Metal Gear games.
I love stealth/action games when they're done well, and AA is a fantastic example of said phenomena. The self-satisfaction that comes from crawling through air ducts, hiding in the shadows, and leaping out onto unaware thugs is just as rewarding as brawling at ten-to-one odds with gadgets and martial arts. The combat and stealth sections are neatly separated, but link together smoothly (sort of). Apart from that, the progression is nicely paced, the upgrades diverse and fun, and the atmosphere of Arkham Island is dense, but not overwhelming.
The story is pretty typical of Batman narratives: one or more supervillains get loose because of some nefarious scheme enacted by one or more of them with the end goal of killing Bats and taking over Gotham in a flood of mayhem. Structurally, it's plain, but within the context of the location of Arkham Asylum, the old formula suddenly takes a new turn. The mindset of Batman's criminal counterparts has rarely been explored in such detail outside the better of the graphics novels.
I would recommend Arkham Asylum to any gamer that has even the slightest non-casual desire in either Batman or "action-adventure." You won't find a more compelling Batman game, and possibly comic book game, or even stealth/action game, at least until Arkham Asylum 2 comes out.
Game Traits applied to Batman: Arkham Asylum (PC) by matrixrunner