They do not have to be subtitled, although that probably helps. I’m basically referring to films made outside of the USA.
Here’s a partial list:
**Jaques de Floret
AND
Manon des Source
Le Cage aux Folles
Dr. Redbeard (Kurosawa)
Throne of Blood (Kurosawa)
Seven Samurai
Babette’s Feast
Educating Rita
Solaris
Satyricon **(Fellini)
I am awaiting delivery of Hiroshima mon Amour. All hail Criterion!
Just picked up La Femme Nikita. Does *Withnail and I * count? And for some quality shoot’n an’ a kill’n, Hard Boiled, although I prefer The Killer, 'cause I’m just a romantic kind of gal 
The Unbearable Lightness of Being
Wong Kar-wai’s “Chungking Express”
Zenster,
I’ve seen dozens of foreign films over the years, not to the point of being a real fan of them per se, but appreciative of the different techniques and styles they have exhibited.
Japanese:
Rashomon
Ran
Seven Samurai
(some other Kurasawa movies whose titles slip me at the moment)
English:
Dead Of Night (an oldie – 1946)
Last Orders
Loneliness Of The Long Distance Runner
French:
Amelie
La Femme Nikita
I know there are Italian and Mexican films I’ve enjoyed, but can’t place titles offhand. Same with Chinese and Indian ones.
As a general rule I prefer the subtitled ones as opposed to the dubbed ones.
I may make a point of scoring IMDB for others and get back to you later. Surely I’ve omitted some I would prefer over those listed, but this is a from-the-hip reply.
The Virgin Spring.
http://us.imdb.com/Title?0053976
While you’re here, what’s the name of the Bergman film where the Knight plays chess with the Devil for the lives of the villagers?
Do you mean “Jean de Florette”? 
Wonderful movie by the way.
I believe it’s “The Seventh Seal” but could be wrong. I did see it anyway.
Oh yeah, thanks!
Well, lesseee…
Citizen Kane, Star Wars, Jaws, Midnight Cowboy…
Uh?
You mean non-US films?
Tampopo
Cinema Paradiso
Everything from the Ealing Studios
My Father’s Glory/My Mother’s Castle
My Life in Pink
The Seven Samurai
The Hidden Fortress
Throne of Blood
Black Narcissus
I’ve seen a bit of Jean de Florette, Zenster, and mean to rent it as well as Manon des Sources soon. The books from which these movies come, by Marcel Pagnol, are very very high on my list of favorite books of all time, BTW.
I’ve always been a huge fan of non-Hollywood stuff but until the advent of DVD, the selection of foreign films in my area has always been pitiful. In the 5 years since I bought a DVD player (early-adopter), I’ve practically over-dosed on foreign films. Here’s a partial list of favourites:
Japanese[ul][li]Tampopo, A Taxing Woman dir. Juzo Itami[/li][li]Hidden Fortress, Ran, Rashomon dir. Akira Kurosawa[/li][li]After Life dir. Hirokazu Koreeda[/li][li]Rampo dir. Rintaro Mayuzumi & Kazuyoshi Okuyama[/li][li]Princess Mononoke, Laputa: Castle in the Sky, Spirited Away dir. Hayao Miyazaki[/li][li]Hana-Bi, Brother dir. Takeshi Kitano[/li][li]Ringu dir. Hideo Nakata[/li][li]Kairo, Kyua dir. Kiyoshi Kurosawa[/ul][/li]Hong Kong/Chinese[ul][li]Chungking Express, In The Mood For Love dir. Wong Kar-wai[/li][li]Bullet In The Head, Hard Boiled, The Killer dir. John Woo[/li][li]King of Masks dir. Tian-Ming Wu[/li][li]Shanghai Triad, Raise the Red Lantern, Story of Qiu Ju dir. Zhang Yimou [/li][li]Emporer and the Assassin, Temptress Moon dir. Chen Kaige[/li][li]The Eye dir. Pang Bros.[/li][li]The 5 Deadly Venoms, Chinese Ghost Story. And a whole slew of other Kung Fu/Wu Xia movies.[/ul]French[ul][]Amelie, City of Lost Children, Delicatessen dir. Jean-Pierre Jeunet[/li][li]Mon Oncle, M. Hulot’s Holiday, Playtime dir. Jacques Tati[/li][li]Rififfi dir. Jules Dassin[/li][li]Wages of Fear, Les Diaboliques, Le Courbot dir. Henri Georges Clouzot[/li][li]The 400 Blows, Day For Night dir. Francois Truffaut[/li][li]La Ceremonie dir. Claude Chabrol[/li][li]Diva, Betty Blue dir. Jean-Jacques Beineix[/ul]German[ul][]Aguirre: The Wrath of God, Fitzcarraldo, Nosferatu, Little Dieter Needs to Fly dir. Werner Herzog[/li][li]Wings of Desire, Paris, Texas, Buena Vista Social Club dir. Wim Wenders[/li][li]Ali: Fear Eats the Soul dir. Rainer Fassbinder[/li]Run, Lola, Run, The Princess and the Warrior dir. Tom Tykwer.[/ul]Man, I think I’ve run out of steam already and I’ve missed a huge bunch of stuff like Leningrad Cowboys Go America, Solaris, The Kingdom, Kristof Kieslowski, Italian Giallo (sp?) horror movies, England’s Ealing comedies, and practically everything David Cronenberg has ever put to film.
Les Vampires (France, 1915)
Viktor und Viktoria (Germany, 1933)
Der Blau Engel (Germany, 1930)
La Dolce Vita (Italy, 1960)
Cavalcade (UK, 1933)
Genevieve (UK, 1952)
Different from the Others (Germany, 1919)
Ma Vie en Rose (France, 1997)
Australian
Breaker Morant
Rabbit Proof Fence
British
Chinese/Hong Kong
Shaolin Soccer
God of Gamblers
Street Fighter
Japanese
Italian
Amarcord (Italian)
Dogs in Space (Australian)
Babette’s Feast is great, too, but I can’t recall what country…
OK, lets try these;
French - Diva, Alphaville
Italian - Life Is Beautiful
Hong Kong/Chinese - Crouching Tiger Hidden Dragon, Hard Boiled, The Killer, The Bride With White Hair, Shaolin Soccer
Japanese - Tetsuo: The Iron Man, Visitor Q, Seven Samurai, Ichi The Killer, Dead Or Alive, Ran, Rashamon, Battle Royale
Spanish - Tie Me Up! Tie Me Down!, High Heels
The Pillow Book would be up there. Amelie came to mind quickly.
Drunken Master II (though that satisfies me on a non-cinematic level)
I’m surprised no one’s mentioned the Oscar winning La Vita è bella (1997) (Life is beautiful), directed by that hilarious chap, Roberto Benigni.
Anyway, mine are mostly Spanish:
Amores Perros (2000) (Love’s a bitch)
Abre los ojos (1997) (Open your eyes - the original ‘Vanilla Sky’)
Hable con ella (2002) (Talk to her)
Todo sobre mi madre (1999) (All about my mother)
Y tu mamá también (2001) (And your mother too)
And besides Amelie, another recent French movie that sticks out is Ma femme est une actrice (My wife is an actress)
I wish I could find The Seven Samurai here, but I’m starting to run out of hope…
Oops… a quick scan reveals that sirtonyh did mention it.
How could I have forgotten One Night the Moon ?