The Maltese Falcon
Unforgiven
Dead Poets Society
Chinatown

The Shootist
From Here to Eternity
Roman Holiday
The Sting
The Postman Always Rings Twice
The African Queen
Young Frankenstein
and my favorite, My Fair Lady
and Ponette if you want to cry.

I’ve seen 38 of the 46 movies on your list, so even though they’re not all my favorites, I can see we have a lot of likes in common.

I’d suggest looking up some more of Akira Kurosawa’s films (in addition to Rashomon), especially these:

Ikiru
The Seven Samurai
The Men Who Tread on the Tiger’s Tail
Drunken Angel
Stray Dog

Also add:

La Strada (Fellini; Quinn, Masina; 1954)
Marie Antoinette (Power, Shearer; 1938)
A Tale of Two Cities (Colman; 1935)
A Christmas Carol (Sim; 1951)

Treasure Island (1950 film)

Up the Down Staircase

Who is Afraid of Virginia Woolf ?

The Collector

Elvira Madigan

Oklahoma in TODD-AO saw that in Los Angeles as a kid.

Rear Window (1954)

Make that 40.

You need a few Ingmar Bergmans on your list; for example,

Wild Strawberries
Through a Glass Darkly
The Seventh Seal

The Mark of Zoro 1920 silent film.

Tom Jones
Cinema Paradiso
The Night of Shooting Stars
Zorba the Greek
Z
Harvey
The Thin Man
A Night at the Opera or other Marx Bros
Nights of Cabiria
Jules et Jim
Breathless
The Discreet Charm of the Bourgeousie
Belle du Jour
The Hunchback of Notre Dame (Lon Cheney)
The 39 Steps
Vertigo

Rebecca by Hitchcock.

The Rules of the Game
Lawrence of Arabia
The Bridge on the River Kwai
Henry V (Olivier)
Richard III (Olivier)
The Wild Bunch
The Americanization of Emily
5 Easy Pieces
The Duelists
Yojimbo
Ben Hur
The Lion in Winter
Paper Moon
Cleopatra

A Man for all Seasons.

Viridiana by Bunuel
Solaris and Stalker by Tarkovsky
High and Low by Kurosawa
Tokyo Story by Ozu

Full marks for everyone on excellent suggestions!
Especially @Jeruba for Stray Dog and @mazingerz88 for High and Low.

I will add if you like Casablanca, watch Saint Jack.

To Sir With Love

North by Northwest

Little Big Man

Butch Cassidy and The Sundance Kid

To Kill a Mockingbird

Nosferato. It was banned for plagiarism 100 years ago. Should be legal to watch now.

Portrait of Jennie, a film from 1948. It’s a hidden, little known gem. A must see, really.

Hell in the Pacific
Duel
Rope
The Defiant Ones
Strangers on a Train
The Light at the Edge of the World
Theatre of Blood
White Heat
Get Carter

I Know Where I’m Going with Wendy Hiller is a delight

The Spiral Staircase
What’s Up Pussycat
West Side Story
The Birds
What Happened to Baby Jane?
Some Like It Hot
Spartacus
Ben-Hur
Miracle on 34th St.
The Odd Couple
Invasion of the Body Snatchers
Its a Wonderful Life
Cool Hand Luke
Anatomy of a Murder
Gone With The Wind
It Happened One Night
Auntie Mame
It’s a Mad, Mad, Mad, Mad World
Hopscotch (though it is from the 80’s)
The Lodger
The Trouble With Harry

Thanks everyone for the recommendations! Will definitely be checking out a lot of these.

‘Hidden Fortress’ (1958, dir: Akira Kurosawa)
‘The Seventh Seal’ (1957, dir: Ingmar Bergman, star: Max Von Sydow)
‘The Conversation’ (1974, dir: Francis Ford Coppola, star: Gene Hackman, Harrison Ford)

@filmfann Not sure why but High and Low seems to be my favorite Kurosawa film.

There’s one about the residents of an insane asylum, where they’re fed nothing but North African pasta: One Flew Over the Couscous Nest. ;-p

Youtube is a great place to find hidden gems for free.

Dr Zhivago
A Star Is Born (the very first one)
The Wizard of Oz
All About Eve
Psycho
The Birds
The Philadelphia Story
Seven Samurai
North By Northwest
Lawrence of Arabia
A Streetcar Named Desire
Chinatown
Rear Window
On The Waterfront
12 Angry Men
The Treasure of Sierra Madre
The Big Sleep
His Girl Friday
Gone With The Wind
Vertigo
The Grapes Of Wrath
Cool Hand Luke
Scarface (the original)
All Quiet On The Western Front
The African Queen
Miracle on 34th Street
One Flew Over The Cuckoo’s Nest
Rio Bravo
Bringing Up Baby
Rebel Without A Cause
West Side Story

I know there’s more but I’m drawing a blank here

And if you like Wendy Hiller (I do) and the dashingly versatile Leslie Howard, put 1938’s Pygmalion on your list. This is the original Shaw play that later morphed into My Fair Lady.