Enjoy many of the above-named films.
Absolute fave: Soldier of Orange (Dutch)
In addition to many of those mentioned:
La Belle et La Bete by Cocteau – the most poetic film ever made
La Grande Illusion – one of the greates of all antiwar films
Skammen (Shame) by Ingmar Bergman – his one science fiction film.
För att inte tala om alla dessa kvinnor (usually titled “All these Women,” but I understand a better translation is “Let’s not talk about all these women”) by Ingmar Bergman. His one slapstick comedy
Yojimbo (has to be mentioned) – the best western ever made 
La Strada – For Guilietta Masina alone
Le Roi de coeur (King of Hearts) – decent antiwar comedy
Le Grand blond avec une chaussure noire (The Tall Blond Man with One Black Shoe) – terrific slapstick comedy
El – Louis Bunuel story of sexual obession.
- Le Roi Qui Danse* is an excellent and beautifully made film about the relationship between the composer Lully and King Louis XIV (the Sun King) - a visual feast to use a critic’s cliché.
There are so many other non-Hollywood films I like that it’s probably pointless to try and list them but singled that one out for special mention. Also Cinema Paradiso and A Room with a View.
A sampling of my favorites, by director:
Fellini:
[ul]
[li]La Dolce Vita[/li][li]8 1/2[/li][li]La Strada[/li][/ul]
Bergman:
[ul]
[li]Seventh Seal[/li][li]Wild Strawberries[/li][li]Fanny and Alexander[/li][/ul]
Kieslowski:
[ul]
[li]The Double Life of Veronique[/li]Three Colours trilogy (esp. Blue)[/ul]
Heh, Mad Max?
My top alltime favorite films (hardcore elitist filmgeek here) are all foreign films. The top spot on my list tends to rotate films like Carl Dreyer’s The Passion of Joan of Arc and Andrei Tarkovsky’s The Sacrifice and Solaris.
Most of my favorites have already been mentioned, but I need to add the Korean movies I like:
Take Care of My Cat
A Bungee Jumping of Their Own
not a fancy art film, but a great comedy: Just Do It
A charming little Danish film; Italian for Beginners.
And speaking of Danish Films, so was “Babette’s Feast”, Fairblue. I really want to see, “It’s a Beautiful Life” and “Amelie,” as I’ve heard nothing but praise for them.
I’m glad to see people mentioning some Australian work here. The “Mad Max” series certainly established a definite niche replete with its own quasitechnical Gibsonesque futureworld. In that vein, I’d like to mention “The Shiralee” starring Brian Brown. It’s a moving story with some rather decent acting. As “Sea Biscuit” hits the stateside screens, I’d also like to suggest another fine equine tale. Simon Wincer’s “Phar Lap” is as good a horse story as anyone can hope for. Another fine Aussie flick that rarely sees the light of day 'round these parts.
Some other great Korean films:
My Sassy Girl
Failan
Christmas in August
The Way Home
Friend
Peppermint Candy
Il Mare
Joint Security Area
Attack the Gas Station!
I second Amélie, Hard Boiled and Life Is Beautiful. Staying in the Benigni theme, I would also nominate Johnny Stecchino (hilarious). And I’m surprised that no one has mentioned Il Postino, a wonderful movie (which is both funny and sad without the one ruining the other). And Waking Ned Devine is pretty good too.
“A Better Tomorrow” -Chow Yun-Fat
“The Killer” -Chow Yun-Fat
“The Proffessional” -Jean Reno
Whew! Where to begin? Some all-time faves include:
The Italian Straw Hat (Clair)
The Lighthouse Keepers (Gremillon)
The Crime of Monsieur Lange (Renoir)
Children of Paradise (Carne)
The Earrings of Madame de… (Ophuls)
Hiroshima, Mon Amour (Resnais)
Contempt (Godard)
The Umbrellas of Cherbourg (Demy)
Au Hasard Balthazar (Bresson)
Playtime (Tati)
Chloe in the Afternoon (Rohmer)
Two English Girls (Truffaut)
La Belle Noiseuse (Rivette)
The City of Lost Children (Jeunet/Caro)
Man With a Movie Camera (Vertov)
Shadows of Our Forgotten Ancestors (Paradzhanov)
Andrei Rublev (Tarkovsky)
Come and See (Klimov)
Ordet (Dreyer)
A Page of Madness (Kinugasa)
I Was Born, But… (Ozu)
Ugetsu Monogatari (Mizoguchi)
The Seven Samurai (Kurosawa)
Branded to Kill (Suzuki)
Red Sorghum (Zhang)
In the Mood for Love (Wong)
Spirited Away (Miyazaki)
The Big City (Ray)
Shoeshine (De Sica)
Voyage to Italy (Rossellini)
La Dolce Vita (Fellini)
L’Eclisse (Antonioni)
Fist in the Pocket (Bellocchio)
The Gospel According to St. Matthew (Pasolini)
Pandora’s Box (Pabst)
Aguirre: The Wrath of God (Herzog)
The Bitter Tears of Petra van Kant (Fassbinder)
Kings of the Road (Wenders)
Ashes and Diamonds (Wajda)
The Double Life of Veronique (Kieslowski)
El (Bunuel)
I Am Cuba (Kalatozov)
The Green Wall (Godoy)
The Spirit of the Beehive (Erice)
What Have I Done to Deserve This? (Almodovar)
Vacas (Medem)
The Travelling Players (Angelopolous)
Persona (Bergman)
Underground (Kusturica)
A Moment of Innocence (Makhmalbaf)
Xala (Sembene)
The Battle of Algiers (Pontecorvo)
La Promesse (Dardenne/Dardenne)
Yol (Goren)
Oh my, how could I have forgotten that one? HILARIOUS! Great movie!
Has anyone seen 8 Femmes? It got wonderful reviews and we meant to see it when it came out in theaters but managed to miss it. I was thinking of renting it – worth seeing?