An alien is on the run in America. To get his kicks, it kills anything that gets in its way, and uses the body as a new hiding place. This alien has a goal in life; power. Hotly pursued by another alien (who's borrowed the body of a dead FBI agent), lots of innocent people die in the chase.

An anthropologist who is part of an arctic exploration team discovers the body of a prehistoric man who is still alive. He must then decide what to do with the prehistoric man and he finds himself defending the creature from those that want to dissect it in the name of science.

The Apollo 3 crew are training when it is discovered that the Russians plan a moon landing. The Americans enact a makeshift plan to land a man on the moon first, using an older style Gemini spacecraft. Lee is chosen as the astronaut instead of Chiz, who was trained for the mission, because Lee has no military connection. Lee has three weeks to train before take-off, and will have to stay on the moon in a shelter for about a year, until an Apollo is ready to pick him up. However the Russians take off two days earlier than expected.

The Americans and the Russians each have a two-person base on the moon. The Americans have had to keep replacing their astronaut teams because they quickly go crazy; they have been using only male astronauts on the unspoken assumption that this would avoid any possibility of impropriety. The Russians, as godless Communists, are under no such constraints, and their male-female team has remained well-adjusted. At the start of the film, a male and female American astronaut team is sent up to replace the sex-starved all-male team. The government insists on them being married first to preserve morality. Most of the story revolves around the eventual consummation of this marriage of convenience, and around their relationship with their Russian neighbors, who keep casually dropping by.