The film has four distinct episodes. In the first, The Dawn of Man, we follow the fortunes of a tribe of ape-like prehumanoids. (Though this seems to have been filmed in the Sahara or some other desert, it was in fact all made in studios in Borehamwood: a revolutionary technique of projection was used for the desert landscapes.) One day they discover a mysterious black monolith, and we realise that this object has given the apes the ability to use tools.
In a famous cut, Kubrick then jumps to the year 2001. Dr. Heywood Floyd is travelling to the moon. In an encounter with some Russian scientists, we learn that a story has spread that a moon base has been troubled by an epidemic. However, this is soon revealed as a cover story. The truth is stranger still: the scientists at the base have discovered an object on the moon which was deliberately buried: four million years ago. And when we see the object for the first time we recognise it (unlike Floyd and the other scientists) as a monolith identical to that which the apes found. When the sun hits the monolith for the first time in four million years it lets out an ear-piercing radio signal....
.... and we cut again, to the third episode: the Jupiter Mission. Five scientists - three in hibernation - are on the spacecraft Discovery, the first mission to the giant planet. (Incidentally, in the original script the destination was Saturn, but Kubrick's special effects team couldn't get the rings to look right!) The two active crew members, Dave Bowman and Frank Poole, are accompanied only by HAL, the computer with complete control over the ship. HAL is meant to be perfect, but it slowly becomes apparent that something is wrong. Poole and Bowman secretly discuss turning HAL off, but unknown to them, the computer is lipreading them while they do so:
To defend himself, HAL first blasts Poole into space, then cuts the life support of the three crew members in hibernation. But Bowman, though he is at first stranded outside the Discovery, manages to re-enter the vessel. In a famous sequence, and one which evokes real (and contradictory) sympathy for both HAL and Bowman, the astronaut enters HAL's memory core and turns the computer off:
As he does so, an automatic procedure is activated which reveals to Bowman (and the audience) the purpose of the mission. The Discovery is following the radio signal, transmitted from the moon, back to its source. In the final episode, entitled Jupiter and Beyond the Infinite, Bowman reaches that source: another monolith, floating around the gas giant. In his small pod, Bowman flies towards the monolith: and into it. In a sequence which simply cannot be described in words, he is sucked through - what? A wormhole or black hole? Another dimension? It doesn't matter... eventually he ends up in a strange room in which he sees himself - or experiences - rapid ageing, until finally, the monolith appears again.
The final moments of the film are the most enigmatic of all. Bowman is gone, turned into a "Star child", a half-human, half-alien being that we see seemingly returning to Earth - but for what purpose, we never find out...