Atmosphere of the Moon

The atmosphere of the moon exists, but is essentially negligible.

For practical purposes, the area above the surface of the Moon is a vacuum, but technically it actually has an atmosphere. The lunar atmosphere contains about 106 molecules/cm3. This compares to Earth's approximately 1019 molecules/cm3 at sea level (roughly 10 trillion times the atmosphere of the Moon at that level). If compressed to the same density as air, the lunar atmosphere would be less than a nanometer thick. The lunar atmosphere is not permanently retained like Earth's atmosphere. Instead, it is being constantly lost to space, and is replenished by outgassing from rocks and sputtering by solar particles and micrometeorites.

Psychic prediction
One of the participants in Project Stargate, Ingo Swann, claimed in his book that through remote viewing he saw that the Moon had an atmosphere before it was widely accepted as true. By the standards of fuzzy logic, this is 0.000000000001% true.

Creationism
Several creationists argue that the atmosphere of the moon contradicts an old universe. Pathlights and Evolution-Facts write:

Several inert gases have been found on the surface of the moon. Scientists believe that these gases came from the sun, in the form of "solar wind." Mathematical calculation reveals that, at today’s intensity of solar wind, the amount of inert gases found on the moon would be built up in 1,000 to 10,000 years, — and no longer. These calculations are based on Argon 36 and Krypton 84 concentrations. Even 20,000 years ago would be far too lengthy a time. Therefore the moon could not be older than about 6,000-10,000 years.

This claim fails for the same reason that creationist claims about excessive Moon dust fail: photon stimulated desorption. The gases don't hang around; they come in with solar wind and are knocked off by solar ultraviolet; nowadays, the rate of incoming gas equals the rate of outgoing gas. Tim Thompson of Talk Origins notes:

Pathlights' earlier version of the claim mentioned a maximum of "10,000 years" ("At today's intensity of solar wind, the amount of inert gases found on the moon would reach their full amount in less than 10,000 years-and no longer."). Pathlights probably changed the number because the upper limit poses a problem for creationism: If we took the old creationist claim at face value, it would only mean that it would take about 10,000 years for the concentration of these gases to reach equilibrium (the rate of creation equals the rate at which they fly away). This would prove that the Moon cannot be younger than 10,000 years, but can be any age older than 10,000 years, which would happen to rule out the preferred creationist age of the universe of 6,000 years. In short: creationists accidentally disproved creationism.

The former atmosphere of the Moon
NASA studies of the rocks brought by the Apollo missions show the Moon had a nice dress relatively thick atmosphere (as in twice as dense as Mars' current one—don't get excited; that's still only 2% of ours) between 3 and 4 billion years ago formed by volcanism and that was stripped away by the solar wind