Keir Dullea, Gary Lockwood, and Stanley Kubrick (Time & Life Pictures/Getty Images/Dmitri Kessel)

Forty-five years ago today, the Uptown Theater in Cleveland Park hosted the world premiere of what soon became recognized as the greatest science fiction film of all time, and one of the flat-out best examples of the entire medium.

On April 2, 1968, Stanley Kubrick’s 2001: A Space Odyssey played for the first time in front of a live audience, projecting its symphonic, harrowing, and beautiful vision of humanity’s origins and destinations on the Uptown’s immersive 70-foot-by-32-foot screen.

Everyone knows the story by now: Prehistoric hominids rise thanks to the power of the Black Monolith; millennia later, space-faring humans discover another device floating out near Jupiter; spacecraft operating system goes insane; Dave Bowman reaches Jupiter, and the stars align into what even after all these years is still one of the greatest achievements in special effects.

“I believe all this is perfectly possible,” Arthur C. Clarke, who wrote the novel of 2001 and co-wrote the screenplay, told The Washington Post a few days before the premiere. “I believe the earth is just a stepping stone. I believe we have to convey to people the excitement and beauty of space.”

And that the film did, and continues to do today. Released a year before the moon landing, 2001 posits some developments that have never come to pass—a lunar colony, for instance—but many others that have. Dr. Heywood Floyd (William Sylvester), on his shuttle to the moon, toys around with a tablet computer.

In honor of the 45th anniversary of the film’s debut, LIFE.com has a gallery of Kubrick and his actors, Keir Dullea and Gary Lockwood, on the set for the Discovery, the HAL-powered spaceship that carried the fatal mission to Jupiter. A few of the photos, taken by Dmitri Kessel, are posted above, while more can be seen at LIFE.com.

And if you really want to geek out, you can always fast forward 2001 to the screen reading “Jupiter and Beyond the Infinite,” put the second side of Pink Floyd’s Meddle on your turntable, and play them simultaneously. “Echoes,” Pink Floyd’s 23-minute-31-second epic, is as long as it takes Dave Bowman to go from star gate to space baby.