This is just a selection of notes, rather than a full reply to every suggestion made! I've looked at all offered, and looked them up to make my list of what to watch, not left anyone out, even if not replied specifically to the suggestions...

I've seen Cowboy Bebop in places. Never anything that struck me (what I'd think) as watchable...

Death Note Well, I saw the two Live-Action films, and quite enjoyed them. Adding to my list!

Kara no Kyoukai Sounds interesting. I'll try (almost) anything once...

Fate/Zero I liked the live-action Battle Royale films.

Ghost In The Shell Yeah, I got one on DVD stashed somewhere in a cupboard. Not bad.

3x3 Eyes This was one of the shows on Ch4 I mentioned. I remember it being good, but would be hard-pressed to give details, being two decades ago, or more!

Akira Movie, right? Think I saw it, many years ago. Post-Apocalyptic? I'll give it a go.

Jigoku Shouju Looks interesting.

Elfen Lied Yeah, seen this one, and it was quite enjoyable.

Ninja Scroll This was actually one of the first, if not the first, animé I bought! Really liked it.

Princess Mononoke? That's the one about forest spirits? It was on TV recently as part of a season of Japanese films. Studio...Ghibli? Gibhli? something. OKAY, don't shoot me!

The only one I watch is "Space Dandy".

He's a dandy guy, in space.

Space Dandy is funny shit but i can't watch more than 2 episodes in a row. :D

*click* *click* *BANG*
I'd give some recommendations, but everything I'd add is already here (except for maybe Trinity Blood. Can't go wrong with almost anything here though.) Maybe Durarara!!, but I only have seen/read the first portion and about half of it is a mystery to me. I like Attack on Titan, don't think I'd necessarily call it adult, more of just dark and brooding.

Yeah, you can't go wrong with AoT, especially if you like cannibalistic giants! :D

Trinity + Blood reminds me to mention Black Blood Brothers as well, while I'm at it.

Besides the ones listed, how about:

Shiki
Steins Gate
Nana
Gankutsuou
Jormungand
Shinsekai yori
Higashi no Eden
Rainbow: Nisha Rokubou no Shichinin
Ga-Rei: Zero
Phantom: Requiem for the Phantom
Gungrave

umm let's see random stuff off the top of my head

Sword of the Stranger
Gungrave
Serial Experiments Lain
Haibane Renmei
Texhnolyze
Michiko and Hatchin
Legend of the Galactic Heroes
Monster
Guin Saga (well, it's kinda pulpy)
Welcome to the NHK
Mind Game
Kaiba
The Tatami Galaxy
Ping Pong
The Eccentric Family
Mononoke (not to be confused with the Ghibli one)
Le Chevalier D'eon
Birdy the Mighty Decode
Casshern Sins (these last two are reboots of older series)
Shiki
Revolutionary Girl Utena
Penguindrum
Kino's Journey
Boogiepop Phantom (though it makes more sense if you know the background of Boogiepop and Others)
Princess Jellyfish
Black Lagoon
Spice and Wolf
Mushishi
Blast of the Tempest
Now and Then, Here and There

The garden of sinners.

"I'll rip your head off and shit down your neck!" - Duke 3D

What makes this quote especially memorable is that he was actually doing it. ;-)

Edit: Oops, that was supposed to go in the game quotes thread. Sorry.

I agree that at the beginning Cowboy Bebop doesn't look that interesting but you should give it another chance and watch it until episode 5: Balled of Fallen Angels, that is when the over arching plot of the series first reveals itself and what got me hooked on the show. Spike is just a really great character and one of the most memorable ones for me from all the anime I have watched.

Also Princess Mononoke is indeed a Studio Ghibli movie and is one of the best by Hayao Miyazaki, it has a more mature tone then the rest of the Ghibli movies so do check it out as well. Some more anime movies that you might like that have a mature tone are Perfect Blue and Tokyo Godfathers by Satoshi Kon. Apart from that I definitely agree with the following recommendations:

Black Lagoon
Death Note
Ghost in the Shell
Gungrave
Hellsing Ultimate
Darker Than Black

Also if you liked Ninja Scroll then I would also suggest Rurouni Kenshin: Trust & Betrayal samurai anime which is a really great adultish OVA series and a prequel to the less mature Ruronuni Kenshin series, if you haven't watched it before. One series I would like to mention that hasn't been listed yet is Ergo Proxy which is a dark cyber punk influenced sci-fi anime.

I almost forgot: The two 'When they cry' series are definitely worth watching, too. Look for Higurashi no Naku Koro ni (When the Cicadas cry) and Umineko no Naku Koro ni (When the Seagulls cry).