Since Cowboy Bebop and Last Exile have already been mentioned, I think some Ghibli is worth recommendation.
Specifically, Nausicaa of the Valley of the Wind
is excellent, the manga even more so than the film. (The movie only covers roughly #1 of the 4-volume series.) The short version: semi-medieval/industrial-tech world plagued by the Sea of Corruption, a malignant fast-expanding forest of spores and massive insects and other fatal characteristics, that threatens to consume the world if humans don't destroy each other first. War is breaking out between the two major kingdoms of the world and smaller nations - such as the village of the Wind, where Nausicaa is daughter of the chief - are getting caught up in the conflict.
A great story, that, but if you'd prefer to stick to more "classic" action-adventure fare, you could do worse than Scrapped Princess
, a single-season (26 ep.) series about a cast-out princess who was meant to have been killed due to prophecy noting her existence would mean the end of the world when she turned eighteen. The execution didn't take. Since then her two "siblings" have been traveling around the country trying to keep her hidden and safe.
On a related theme but with a very different tone (Scrapped Princess involves a good amount of magic and demons and so forth throughout, this series is relatively more down-to-earth up until the last couple of episodes), Moribito
follows female bodyguard Balsa after she agrees to protect the 2nd prince, Chagum, from his father. The boy is possessed by a water spirit which, according to the royal stories, can only be expelled if the boy is killed by his father, the emperor. The boy's mother can't stand to let it happen and hires Balsa to take him away and hide him. Another single-season series, this one based on the first of a 10-book series of novels (most of which sadly aren't available in English). Sidenote, while Scrapped Princess is very actiony, Moribito is much slower and more peaceful, focusing on character development through conversation and political interactions more than flashy fight sequences (though its few fight scenes are quite well done, and more concise than the usual hack and slash - these people don't fuck around when they mean to kill someone).
Transitioning from feudal Japan to sci-fi we have Samurai 7
, a futuristic re-imagining of Kurosawa's Seven Samurai. Need I say more than that to describe it, really? Take the plot of Seven Samurai, give it some steampunk anime flavor and you're pretty much there.
Last, to complete the shift to sci-fi, Ghost in the Shell
. Any version of it is great, but if you want to go through all of it in roughly "correct" chronological order (story-wise, not when the shows were created), you'd want to start with Ghost in the Shell: Stand Alone Complex, then GitS:SAC 2nd Gig (the second season), then the first movie - which is simply Ghost in the Shell - and then the second movie, GitS: Innocence. Fortunately, you can watch them out of order and it'll still be good 'cause they're ALL good stuff; the inside jokes and references and so forth just make more sense if you watch them in order for character progression. Of course, if you read the manga, you'll notice a lot of overlap as the series and movies borrow from the manga pretty heavily in some areas - the whole first movie is lifted directly out of two chapters of the manga, for instance. Not that that makes the material any less well executed on screen.
Hm...that's all that comes to mind, really. Blade of the Immortal
is a good chop-chop samurai series but it's fairly long (up to volume 23+ now, I think?) and I haven't seen the anime to compare it. I feel like I'm forgetting something though...
Oh, of course! Gunslinger Girl
. Great series. It's about a secret Italian government organization that takes young girls who are in otherwise desperate positions (quadriplegic, victim of a group making a snuff film, contracted bone cancer, etc.), wipes their minds and rebuilds them into cyborg enforcement agents for assassinations, protection work, espionage, etc. under the supervision of their adult handlers who make sure they don't just go nutso and give them direction in their lives, as the Conditioning leaves them rather hollow as individuals. ... It's better than it sounds,
though in this case the manga outstrips the anime by a fair margin in quality and depth. Still, the anime has its moments; Rico's hotel job comes to mind as a great example.
...yeah, I think that's good enough for now.