Action/Adventure:
* Raiders of the Lost Ark
* Red Dawn
* Rambo
* Reservoir Dogs
* Raging Bull
* Rush Hour
* The Revenant
* Rogue One: A Star Wars Story
* Rocky
* Ronin
Comedy:
* Raising Arizona
* Rat Race
* The Royal Tenenbaums
* Rushmore
* Romy and Michele's High School Reunion
* The Rundown
* The Ridiculous 6
Drama:
* Rain Man
* The Revenant
* The Reader
* Revolutionary Road
* Requiem for a Dream
* The Right Stuff
* Room
* Rear Window
* Rebecca
* Rosemary's Baby
* Raging Bull
Horror:
* The Ring
* The Silence of the Lambs
* The Sixth Sense
* REC
* The Ring Two
* Rosemary's Baby
Sci-Fi/Fantasy:
* The Raid: Redemption
* Ready Player One
* The Ring
* The Riddler
* Riddick
* Return of the Jedi
* The Matrix
* The Martian
Animation:
* Ratatouille
* Rio
* Rango
* The Red Turtle
Musical:
* The Rocky Horror Picture Show
This list is just a starting point, and there are many other great movies beginning with "R" out there. I encourage you to explore and find some new favorites!