I like some of the classic (or not the newest at least) shoujo (i.e. girl-oriented) anime so Fushigi Yuugi (Mysterious Play, which my husband also liked). A bit darker by the same animator (Yuu Watase), Ceres: Celestial Legend has the most dysfunctional family I think was ever written.

Escaflowne has stuff for shonen (boys) as well as shoujo, but we both think it has a wtf (not WTF, but slightly wtf) sort of ending. Other than that, we loved it. I recommend Ranma 1/2, especially first couple of seasons, for sheer slapstick. (This may be the inspiration for Kung-Fu Panda, for reasons that will become obvious.)

Silly at first, but going philosophical, is Trigun. (The main character reminds me of everything I liked and half of what I hated about my ex-husband, so it's a bit bittersweet for me). We both loved that.

Isn't Bleach kinda shonen-ai (boy-love-ish?)

Barefoot Gen. It's an account of the artist's childhood when he lived through the nuking of Nagasaki, and is one of the most haunting movies I've ever seen.

Bleach is just shonen - it's serialized in Jump.

Get Cromartie High School RIGHT NOW. It's one of the funniest things ever, whether in anime or manga form (the live-action movie is probably best appreciated by fans).

No, really:

TEKKONKINKREET!

...and Tokyo Godfathers.

And Barefoot Gen, as mentioned by BP.

Must-see classics. Do yourself the favor. Etc.

You might like Foolie-Coolie. Kinda fun, but don't expect much.

See the original Ghost in the Shell.

Not anime, but the animated Renaissance is quite a visual accomplishment.

jtron, I'll check out Cromartie. Thanks!

weretable, please check back in and give us recommendations and tell us what you liked and didn't?

I'm sure I'm missing some things above, but maybe they'll come to me.

/Sorry to be so paragraphy.

Oh, yeah: You'd really better see the legendary Akira, too. Kinda pivotal in the history, ya know. And ya gotta love the motorcycle gangs.

Did I only say "motorcycle gangs"? Juvie delinquent, pill-popping motorcycle gangs rumbling while riding.

That's better.
;-)

I agree with Akira. I recommend Macross, My Neighbor Totoro, Gundam, Evangelion, Oh My Goddess, Kiki's Delivery Service, Kareshi Kanojo no Jijo (Kare Kano), and Battle Angel Alita. I thought the Escaflowne TV series was better than the movie.

Anything made by Miyazaki / Ghibli Studios. Quick recs: Princess Mononoke and Spirited Away are both recent and have excellent English dubs if you're subtitle-averse. And everyone should see Nausicäa of the Valley of the Wind before they die.

Shane: Hmm, I have Tokyo Godfathers, Tekkonkinkreet, and Paprika (which you did not mention, but whatever) sitting in my to-watch pile. Hooray!

I don't watch much anime, but I really liked Planetes (hard SF, sent in near-Earth outerspace) and Serial Experiments Lain (sort of what you'd get if William Gibson and David Lynch collaborated on an anime, I guess). And Giant Robo, just for the animation quality.

I second Ghost nin the shell. I saw it a real movie theatre in San Fran 96 and that at the time, was ossom. It might feel a little dated now but still cool.

I wasn't so fond of Akira. Ok, I was fond of all of it except the overblown pestulous-bleb-tentacles at the end. Is it just me? Probably.

Highly recommend Miyazake/Ghibli as well...

I'm not an anime fan, but I get exposed to it regularly through friends.

For sheer silliness, I'd recommend the two-part series Dragon Half. Lotsa full-out belly laughs.

I saw a couple of episodes of The Last Exile, which story-wise was a bit off, but was one of the most beautiful animated things I've ever seen.

And last and probably least, I was subjected to Interstella 5555 a few months ago. Has to be seen to be believed.

You really can't go wrong with anything put out by Studio Ghibli. Howl's Moving Castle and The Cat Returns are my personal favorites.

I normally like the disturbing/philosophically deep/fuckedupinthehead. The kind that have schizophrenic characters, a schizophrenic reality, and are filmed from a schizophrenic perspective. So you may just hate the weird ones I'm recommending (Lain, Perfect Blue, Paprika). I know all my friends do :).

(my recommendations will be from the time I liked anime, so maybe a bit old)

Series:
Serial Experiments Lain - This is about the internet mixing up with the fabric of reality. It has a unique style... and ever since I watched it I never failed to notice when I walked past a power transformer.
For adult-themed mecha series (Less philosophy, more action), Evangelion, Argent Soma, Rah Xephon. They start to look all the same, though.
If you're looking for romantic comedies, I second Ah! My Goddess. Also, Love Hina and Vandread. They also start to look all the same, and when you get bored be sure to check FLCL (Furikuri) which parodies all the cliches in the above.

Singles:
Perfect Blue and Paprika. These really managed to draw me into the character's insanity. Perfect Blue managed to put me in "third person mode" (observing what happens with me as if I weren't involved) for more than a week.
For less insanity, I second everything from Ghibli. Especially Mononoke.

I wasn't so fond of Akira. Ok, I was fond of all of it except the overblown pestulous-bleb-tentacles at the end. Is it just me? Probably.

Highly recommend Miyazake/Ghibli as well...

No, probably not just you. The ending left me wanting, too, and I can only imagine how much better the ending of the manga was, as I only have maybe 3/4 of the series in comicbook form.

And I agree with everyone recommending Miyazaki/Ghibli.

If you happen across a copy of the collection of shorts Manie Manie, sometimes marketed as Meikyu Monogatari or in English as Neo-Tokyo, I really recommend it as good fun. It has a popular short from the early days of MTV's Liquid TV, as well as one from Akira's Katsuhiro Otomo. If you can't find a copy, everyone just stop by my house someday for an anime party.

jtron and qvantum, thanks for the recommendation on Paprika.

Sometimes anime and cartoons and animation in general are way more fun with the sound turned off. You notice all kinds of things you normally wouldn't and you appreciate the visual aspect much more. Not to mention it's just fun. Once you've viewed something normally, maybe put it back on later with the volume down while you're doing paperwork or housecleaning (or having a party!) Even many series that you normally might not like (and that are nowhere near the level of "serious" stuff like Tekkonkinkreet or Ghibli or most of what we talked about above) become fun with no sound, even series like Dragonball Z (although it's a shameless, ridiculous, overly violent cock-fight), or even StormHawks, currently on CartoonNetwork in the U.S. Mon-Thurs at 10:30p. Even Ben 10 has its moments. (Okay just shoot me, LOL.)

One more: if you get a chance to see the original '60s Astro Boy series, I find it to be good fun. Camp back in the '60s and more so now, but good fun.

Oops: Link to Manie Manie (and a decent info anime source.)

I've never been able to make through any Ghibli movie except Grave of the Fireflies. The few I tried (can't remember which) were all just too sickly sweet for me to stand. So I'll stand bravely against that tide.

If you liked Bebop:

Ghost in the Shell
Appleseed (the newer CGI one is neat)
Akira
Planetes (warning: I only watched the first couple discs)
Neon Genesis Evangelion (warning: depressing. warning: weird.)
Martian Successor Nadesico (lots of jokes about anime)

And Grave of the Fireflies, but hide everything sharp first.

And, DISCLAIMOR: I am not an otaku!! Watashi wa otaku... damn, can't finish the sentence, forgot the negative form. I'm just a general pop-culture-head.
;-)

Oh, and you'll enjoy the Bebop movie if you haven't seen it. It's okay.