Off the top of my head:

Hikaru no Go - anime about a kid learning to play Go with the best in the world. I've only read one of the manga, but it looks pretty good.

One Piece - silly fun about a pirate kid named Luffy who can stretch like rubber. Very fun, very popular, recommended.

Naruto - little more intense (but only a little) about a kid who desperately wants to become a powerful ninja. More violence than the other two, and in general a little more serious.

If your kids like manga (the comic form of anime) all these series are available in stand-alone books and in the magazine Shonen Jump. I'd check out Shonen Jump just to get a feel for other anime they may like, since pretty much all of those manga are also anime.
These are all series, though, so it's a little different than the movie-length ones they've seen.

Alex

Thanks for the tips Alex. I went ahead and subscribed to Shonen Jump after walking through some of the online flash versions - I think the kids will love it.

I LOVE Trigun. There are a few swear words in some of the subtitles, but no nudity problems throughout the whole series. Very funny stuff a lot of the time but a good depth of plot, too. Its an apocalyptic western style. (The main character is a gunslinger who doesn't like to shoot people....)

Cowboy Bebop is pretty good, but a bit darker. Great soundtrack to boot. Some drug trafficking, etc... (the guy is a bounty hunter catching criminals) and a bit of violence here and there.

Both of those are series... you can sometimes find them in the local public library.

The enormously long Macross saga is also pretty good, but not quite on a par with the previous two.

Hi Chris,

My sons Greg and Charlie are both big anime fans. Here are a few suggestions from Greg:

Fruits Basket

Full Moon wo Sagashite (not released in USA, no USA liscence)

Grave of the Fireflies

.hack//SIGN

Haibane Renmei

Hikaru no Go (not released in USA, liscenced by Viz Communications, Inc.)

Martian Successor Nadesico

Vision of Escaflowne