If you're a fan of the slower-paced, more cerebral film, I'd recommend the original Solaris - that way you get Russian and Sci-Fi!

Quoting pauls42:

Quote:
Dark cityv (you need to get the correct region code for your player)
Cloverfield (all region)

I have Cloverfield and by Dark City the one with SUtherland or was there another?

For Russian films you might want to take a look at the web site for the Russian Cinema Council.

Quoting schultzy:

Quote:
Quoting GSyren:

Quote:
For Russian films you might want to take a look at the web site for the Russian Cinema Council.

I saw a couple interesting ones, but looks like they only sell region 5 (believe that is the region Russia falls under).

The 6 titles that I own are all region free. And I seem to remember that most titles could be ordered in either PAL or NTSC, so if the ones you are interested in are available in [whatever standard you prefer] you should be OK.

I hope I've not got too carried away here.  I've intentionally left some stuff out, and even then wonder if I should have put it in.  If you want to source any of these and are having trouble, let me know.  Also, if you like some of my suggestions, look through my collection, because there's even more in there matching your criteria (e.g. left out Tamala 2010: A Punk Cat in Space, Wonderful Days and Storm Riders).

Sci-Fi
I enjoy the Russian movie Stalker (by the same director as Solaris which was suggested earlier).  I can't say I'd recommend it to everyone, because it's loooong, and not heavy on sci-fi elements (it's more that it's a sci-fi concept).

G.O.R.A. is a great Turkish parody of a few sci-fi movies, and stands in it's own right as a top movie.  1984 is a great sci-fi dystopia (and was made in the UK, so counts as "Non-US").  The two recent Appleseed movies from Japan look stunning, and have a well fleshed-out world and stunning visuals.

Nirvana is a pretty cool sci-fi movie with a great range of locations.  It's from Italy, and the Italian release is subtitled.  From what I've read, you want to avoid the R1 release (dubbed), and get this one instead.  I also suggest Avalon, which doesn't appear in my collection for stylistic reasons, but is still a great film.

Pale Cocoon and Voices of a Distant Star are both short (anime) films from Japan.  Pale Cocoon I can't explain much without giving it away, so I'll quote from AniDb.

Quote:
In the far future when the continuity of the history has been lost, it is the world where ruins spread to far away. The sea and the sky have disappeared, and the past scenery exists only in the records excavated in the relic.

The Bureau of Record Excavation is an organization responsible for this task. Ura works for the 92nd office of the bureau. He is absorbed in various records and the world in the past. Riko keeps a little distance to watch him, and she averts her eyes from the records.

Everyone in this world knows that to know the past means to know the misery of this world.

One day, Ura restores a strange video.

Voices of a Distant Star is about a couple where she goes off into a galactic war, and he remains back on Earth.  As she travels further away, they age differently (and there's the time delay).  Quite touching.

Save the Green Planet did amazingly on the festival circuit.  It's sci-fi/dark comedy with elements of horror.  A man kidnaps a CEO of a major corporation, believing him to be an alien from Andromeda, bent on taking over the world, and tortures him for information.

Fantasy
Pan's Labyrinth you've probably heard about already, so I won't say more.  Mirrormask is a great fantasy film with unique visuals.  Some ground already covered by other stories, but original enough to be great fun.  Brotherhood of the Wolf is a dark/horror fantasy from France, with lots of action.

Gormenghast and Neverwhere are both excellent BBC miniseries.  The former is a satire with ancient tradition and a kitchen boy working his way up through the ranks.  The latter is about a wainscot reality below the streets of London, and a man who is drawn into that world (written by the highly talented Neil Gaiman).  I'd also suggest checking out the modern fantasy Jekyll.

Also look into some of the movies from Studio Ghibli.  They're widely regarded.

While we're on the subject, does anyone know if this is available on DVD somewhere in the world?

Casshern (Japan)
http://www.youtube.com/watch?v=JpUWsMzwpAA

Yesterday (Korea)
http://www.youtube.com/watch?v=2MuMAl9o1bo

Resurrection of the Little Match Girl (Korea)
http://www.youtube.com/watch?v=lUVAo60KrM8

Natural City (Korea)
http://www.youtube.com/watch?v=jVRwiMsCP9M

Jade Warrior (Finland / China)
http://www.jadewarrior.net

The last one is fantasy the rest are scifi.
For asian movies i use http://www.yesasia.com