Okay, tin-foil hat time! I have heard quite often that (at least some of) the moon landings were a clever hoax, in order to prove the USA's supremacy in space exploration. It sounds like paranoid malarkey, and yet when you review some of these claims - well, they don't seem quite as readily dismissed as you might first think. I'm not minimizing the brainpower and raw heroism that went into the manned moon landings, but these are fun to think about and are the most commonly cited reasons...
1 - No Stars! None of the photos from the moon's surface seem to show any stars in the sky, even the good quality, higher resolution photos. You would think that they would be blazing, with no atmosphere and a black sky.
2 - Weird Shadows! As already mentioned, the Moon has no atmosphere, so the sunlight is direct and harsh; a single point light source casting solid shadows. And yet...there are photos that appear to show shadows being cast at different angles, suggestive of multiple light sources - like on a movie set. You would expect the shadows to be parallel if light by the Sun alone.
3 - No Sign Of Landing Thrusters! Weirdly, you never seem to see any "blast marks" under the LEMs. You might expect the lunar dust to be blown around, but it's not evident - like the Lunar vehicle was just placed there.
4 - The Flag Waves! Of course, with no atmosphere that should not happen - there is no wind. This is less convincing to me, and the TV Mythbusters even took a crack at this one: