Thanks for the recommendations everyone!! I actually picked up a few today to check out!

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Thanks for the recommendations everyone!! I actually picked up a few today to check out!

Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb
Might edit this to add more as I think of them.
Some I haven't seen mentioned:
Come Drink With Me 1966
Barbarella 1968
An angel does not make love, an angel is love
Yojimbo - samurai action/comedy
Samurai Rebellion - serious samurai (also Mifune)
Young Girls of Rochefort - French musical that will make you smile the whole time
Sword of Doom - reverse samurai where you follow the bad guy (and oh what a performance it is)
Le Samourai - French hit man movie with Alain Delon
The Apartment - one of the best written comedies of all time
Playtime - utterly unique French film that bankrupted the director (much to everyone’s loss)
The Dollars Trilogy - where spaghetti westerns were born and still some of the best music ever put on film
Pale Flower - the most noir I’ve ever seen a film achieve.
8 1/2 - a film about making film and much more
Tokyo Drifter - mediocre story that oozes style and has the incredible use of limited colors.
More:
Dirty Dozen, Butch Cassidy, Mary Poppins, It’s a Mad, Mad, Mad, Mad World, and several others that have already been mentioned.
Besides some of the ones you've already named, here are some of my favs:
The Virgin Spring
The Manchurian Candidate
Rosemary's Baby
Anything Sergio Leone
If I remember correctly the The Manchurian Candidate was delayed out to movie theaters for over a year because of the Kennedy assassination.
I looked this up on Schmopes and while your awful memory had failed you (sigh), the actual story is mildly interesting.
Check it out!
Since you like Psycho, I'd recommend Peeping Tom by Michael Powell or several movies by Mario Bava from the era like The Girl Who Knew Too Much and Blood and Black Lace. Eyes Without a Face is also Great. Also Rosemary's Baby and Repulsion by Polanski. For Japanese Films you should check out Yojimbo, Sword of Doom, and 2 by Seijun Suzuki ; Branded to Kill and Tokyo Drifter. Also check out Dr. Strangelove by Kubrick and if you like New Hollywood westerns , etc Bonnie and Clyde and The Wild Bunch are requisites. French New Wave stuff is important so maybe Breathless and Band of Outsiders and of course Melville's "Le Samourai." Spaghetti Westerns were very influential and Leone's Man With No Name Trilogy ( A Fistful of Dollars, For a few Dollars More, The Good the Band and the Ugly ) and Once Upon A time in The West are a good starting point. There are so many but those are a few recommendations that might send you down further rabbit holes.
In addition to what others have mentioned:
Lawrence of Arabia. Get the 4k remaster if you can, it looks incredible.
Doctor Zhivago
Bullitt
The Producers
The Hustler is an excellent Paul Newman film about billiards.
Blow-Up is a bizarre anxiety drenched film about a photographer who may have captured evidence of a crime in the background of a photo.
Grand Prix is a neat film about formula 1 racing, with some of the best action photography ever.
If queer film is your jam, the work of Kenneth Anger is always worth checking out, Scorpio Rising being my favorite of those I've seen
If you like sci-fi and experimental shorts, Chris Marker's La Jetee is pretty cool, and is what 12 Monkeys was based on.
As for super ambitious metanarrative experiments, Symbiopsychotaciplasm: Take One is incredible.
*La Grande Vadrouille", or by it's English title, "Don't look now, we're being shot at." Three RAF pilots accidentally strand themselves in occupied France. With the help of the resistance, and some guys who were in the wrong place at the wrong time, they try to escape. It's definitely a product of its time, but it's not too bad compared to some other titles of this age. Definitely worth a watch!
One that I haven't seen mentioned yet...
The Graduate.
Wild in the Streets (1968)
> If you're thirty, you're through!
The Haunting is great, if you like scary movies. Also directed by Robert Wise, but very different from The Sound Of Music.
Or you might find it insanely boring (as I do). I’m not convinced Robert Wise could helm a well-paced movie.