The Nintendo DSi runs all DS-compatible games except for those that require peripherals to play. The DSi doesn't have a GBA slot, so it will not play Game Boy Advance games or any DS games that use that slot for anything, e.g., Guitar Hero.
As of this date there are few (if any) games available specifically for the DSi, but there should be a bunch of 'em out this summer. Nintendo didn't bother to roll out a plethora of games for their new system like they usually do because there were so many that would run on the DSi system already.
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