The granddaddy of all space conspiracy theories has to be that the moonlandings were faked on a soundstage. Those who believe the moon landings were hoaxes say that the United States lacked the technology necessary to transport humans to the moon and back. They claim that NASA faked the landings in order to make people believe the U.S. had fulfilled President John F. Kennedy's promise to land a man on the moon before 1970.
What evidence do these skeptics cite? For one thing, on the lunar landing videos, you can't see any stars in the sky. NASA says that's because the moon's surface and the astronauts' suits were so reflective that it was too bright for the camera to pick up the comparatively faint stars. Another point theorists make is that while planting the American flag in lunar soil, the flag appears to wave. With no air in space, how is that possible? NASA says that the astronauts rotated the flag's pole back and forth while planting it so that it would remain upright. The rotation of the pole caused the flag to move back and forth as if rippling in the breeze even though there's no air on the moon.
NASA says that there's plenty of evidence that men landed on the moon. There are photos, videos and testimonials from the dozen astronauts who have set foot on the moon's surface. The astronauts returned with soil and rock samples, which NASA also cites as physical evidence of our presence on the moon. Some hardcore conspiracy theorists maintain that all of this evidence is fake or came from unmanned missions to the moon.
That's the real problem with conspiracy theories in general -- there's no real way to convince people who believe in them that they might be wrong. The theorists may claim that any evidence contradicting their ideas was fabricated in an effort to cover up the truth. They may also argue that the lack of evidence to support their beliefs is due to the government (or some other responsible party) taking great pains to remove all evidence from view. In other words, arguing with some theorists is like saying "heads you win, tails I lose."
But even if these conspiracy theories don't reflect reality, it's no secret that there are boundless mysteries waiting for us out in space. We might even know two or three of them already -- but we aren't telling.