Their theory is that in the 1960's the Americans didn't have the technology to send a man to the moon. However, in order to distract public attention from the attrocities of the Vietnam War, and to demonstrate their technological superiority over the Soviets, the Americans faked the first moon landing in the Arizona Desert. NASA was transformed into a science-fiction movie company - a public relations diversion under which billions of tax-payer's dollars were redirected into secret military projects that the government would otherwise never receive funding for. 

It's an intriguing theory, but unfortunately the best conspiracy theories often have the least supporting evidence, and the faked moon landing theory is no exception. The most frequently cited line of evidence involves the shadows cast by the astronauts. For example, in the image below, the shadow of one astronaut is longer and at a different angle to the shadow of another. To the conspiracy theorists, this implies that the astronauts were illuminated by two independent light sources when the photo was taken. There is only one source of light on the moon, and so therefore the moon footage must have been faked on a set with at least two light sources. But the conspiracy theorists are obviously not very good at geometry. Provided that we assume that the surface of the moon is not perfectly flat, then the differing shadow angles and lengths can easily be explained in terms of shadows being cast across an uneven surface. And if the astronauts really were lit by two light sources, then why doesn't each astronaut have two shadows? 

Another frequently cited line of evidence is that there are no stars in the sky in any of the lunar photographs, such as the one below. The conspiracy theorists think that there should be, but that NASA either forgot to put them in, or found their arrangement too difficult to replicate. But the conspiracy theorists obviously don't know very much about photography either. As any photographer will tell you, it is practically impossible to capture a bright object in the foreground and a dim object in the background in the same exposure. Had they done a long exposure to capture the stars, then the astronaut would have appeared as an overexposed blob in the foreground, and we can only guess at the conspiracy theories this would have inspired. 

Even less convincing is a line of evidence consisting of a series of blobs of light in the moon's sky, also seen in the above photograph. Remarkably, the conspiracy theorists who draw attention to these blobs don't actually explain how they relate to their theory. It is as if the sheer presence of white blobs in the sky immediately confers a conspiracy of some variety. But as it turns out, the blobs can be easily understood as lens flares, which are the result of bright light reflecting off the interior of a camera lens. You could probably even go through the photo album of one of these conspiracy theorists and convince them that, because some of their photos have white blobs and because their shadow is cast differently across uneven surfaces, that their photos are fakes and their memories of these occassions were implanted by post-hypnotic suggestion. 

Perhaps the best evidence that people have actually been to the moon is the presence of a laser reflector that was placed on the moon by the Apollo astronauts. Anyone with a laser pointer and a little persistence can shine a laser beam at the specific location of this reflector and detect the reflected beam with a light collector. How can this be explained if humans never landed on the moon? 

Ironically, a conspiracy theorist with a little persistence and creativity can mould their flexible hoax theory around this fact. And they have done this. There are conspiracy theorists who have proposed that aliens beat the Americans to the moon, and colonised it as a convenient location from which to visit Earth in their UFOs. According to this theory, the only time that people have been to the moon is when they have been escorted there by aliens, and NASA faked the moon landings in an attempt to cover up the existence of extra-terrestrial intelligence from the public. After all, why do you think there hasn't been another publicised moon landing since 1973? 

Well, maybe this is because the political incentive to spend millions more tax-payer's dollars on moon missions has been significantly reduced, because aliens had already landed on the moon by 1969, and the aliens were us.