I'm no anime buff, but I can see that there are a tonne of people within this thread who have ridiculous ideas about what is "mature" and what isn't. Particularly OP....what with the whole classification of Baccano and Bebop as 'silly fun' 'n all.
Although I was at first skeptical about the dude who recommended the first part of Bleach alongside a bunch of other, darker properties, I've got to say that I totally agree. Yeah, you could argue that it is about a young man's psychological development given flesh, so to speak, but that would be wankery of the highest degree. However....that whole first section is really, really restrained and well done, less about the fighting and more about the consequences of the conflict, and, more interestingly, about the consequences of avoiding conflict, coupled with some quiet character development and social commentary.
Personally, I'd recommend FLCL - but only if you are willing to watch it twice. Once you get past the initial hurdle of absolute confusion and pretensions to childishness it becomes really fucking clever.
Or House of Five Leaves - best subtle character development I've seen recently. In anything.