Things to do in New Orleans?

I've only been to New Orleans once, and that was before Katrina--10 years ago or so--and I didn't get to spend much time. My impression was that while the French Quarter has its moments, it's mostly touristy and horrible, and that the surrounding areas are potentially very dangerous. But, obviously, things could be very different now.

What should people not miss in New Olreans?

--Mr. Zero