For a grown-up anime (and by that, I mean you can handle grim stuff), I'd recommend Black Lagoon. It's a Mad House production so the animation is excellent, it's a modern-day story so there's no fantasy or sci-fi, and the story is character-driven and pretty deep sometimes. However, I skip the second season's Vampire Twins eps as they're a) disturbing - VERY disturbing, and b) just not as believably done. They are good from a story point, so that's up to you.
Regarding endings, that show doesn't really. It's still progressing but the manga is coming along slowly, so it'll be awhile before a third season.
I'd also recommend Full Metal Alchemist / FMA Brotherhood. They're both good in different ways, although for me the endings of both were not to my taste. But great story and gotta love the characters.
Others I like include Cowboy Bebop (best depiction of a hand-to-hand fight scene ever in the first episode), Ghost in the Shell Stand Alone Complex 1, Last Exile, Claymore (although my teeth ache to re-edit it for pacing), and Moribito - Guardian of the Spirit.
Oh, and subs. ALWAYS subs. Cowboy Bebop had the only dub I found watchable, and it was still only 3/4 as good as the original voices. Everything else, not so much.
