Sci-fi anime ensues...
Ghost in the Shell movies, the first is a classic, the second gets a little bogged down in philosophy, though it looks beautiful.
Ghost in the Shell: Stand Alone Complex. The TV Series, actually two series and a spin-off movie named Solid State Society. One of my fave anime. action sci-fi with a whole lot under the bonnet, set in a future where the line between man and machine is blurring, as the majority of people are cyborgs.
Planetes - Awesome near future vision of manned space exploration, wrapped up in a slice of life character comedy drama. And yes, there is no sound in space.
Noein - The best extrapolation of quantum mechanics I have seen in any medium. A tale about infinite parallel worlds, told through the eyes of a young girl and her friends in a sleepy seaside town.
Serial Experiments Lain - Mind bending cyberpunk tale, an early extrapolation of where the Internet might have gone, made in the early nineties, so aspects of it are outdated, but it's still a classic.
Texhnolyze - creepy, dying days of civilisation tale, with society stratified and living underground, focusing on a prize fighter named Ichise.
Gilgamesh - the world has to come to terms with a terrorist event that sent it into another depression, but somehow coincided with the advent of children with special powers. Another creepy unsettling tale, with odd, but moody animation.
Wolf's Rain - Yet another dying days of civilisation tale, one of the best anime around. A world running down, where wolves are in legend heralds of the apocalypse, but where they are reputedly extinct.
Kino's Journey - An anthology of tales as a traveller and a talking motorcycle journey from land to land. A brilliant anime that I can't do justice to in a paragraph.
Ergo Proxy - The Japanese sure do a lot of these end of everything, post apocalyptic tales don't they. Not all are as good as this atmospheric tale of man and machine's uneasy coexistence in the distant future.
Full Metal Alchemist - Alternate history where alchemy is a real science set around the turn of the twentieth century. It isn't a pure sci-fi, but has a lot of steampunk styling to it, and while it is aimed at a general audience, it has a great story to tell.
Darker Than Black - The show that Heroes wished it coule be. People with superpowers who have to pay a price for their abilities, used as weapons in a new cold war by the world's superpowers.
Gankutsuou - The classic Count of Monte Cristo tale moved to a future setting and given a very unique look.
Gantz - A weird ball appears in a room, suddenly recently dead people are being resurrected to fight in strange and lethal games against aliens.
Mushi-shi - A wanderer helps people who are afflicted by mushi in early twentieth century Japan. Mushi are the smallest most primitive form of life, smaller than even bacteria and viruses, but have weird effects on those they interact with. An atmospheric and stylish tale of man's coexistence with nature.
FLCL - Mind bending classic animation. Story of a boy who gets hit over the head by a guitar, and a robot climbs out of his forehead.
Great soundtrack
Starship Operators - A pretty fair space opera, with a realistic bent when it comes to space warfare.
Eden of the East - Brilliant, brilliant cyberpunk tale, to tell more would be spoiling it. It's a great mystery tale with wonderful characters that is a top recommendation.
Akira - The cyberpunk classic movie that really kickstarted all this anime nonsense.
Metropolis - Godfather of anime Osamu Tezuka reputedly only saw the poster for Fritz Lang's movie, but that was enough to inspire this tale, which was adapted into a film after his death.
Paprika - Sort of beats Inception to the punch by five years.
The Girl Who Leapt Through Time - the clue's in the title. Amazing teen coming of age story.
The same director made Summer Wars, also a film well worth watching.
Sky Crawlers - Beautiful and deep tale from the maker of the Ghost in the Shell films, about children fighting an aerial war for no reason anyone can explain, the tale of a new pilot who tries to find out what happened to his predecessor.
The Wings of Honneamise - My favourite anime movie of all, an alternate history of the Space Race, one of the most beautful, detailed anime ever made.
Other titles worth a look
Witchblade (yeah, they remade it)
Tenchi Muyo (classic)
Samurai 7 (futuristic retelling of Kurosawa classic)
Shingu
Speed Grapher
Utawarerumono
Witch Hunter Robin
Rin: Daughters of Mnemosyne - Immortals! Better than Highlander!
Baccano! - Immortals, but during prohibition!
Cyber City Oedo 808
AD Police
And I would highly recommend Cowboy Bebop. It's brilliant storytelling, even if it is getting on a bit in years. At least try the Cowboy Bebop movie, that's a standalone tale that doesn't need the series to be enjoyed.