Abstract The properties of a neutral lunar atmosphere are investigated theoretically. A non-uniformity is shown to result from the temperature variations and non-uniform gas source distribution on the surface of the Moon. An integral equation governing the distribution of molecular fluxes, in the steady state, is formulated. This equation is solved by computer and analytical methods. Solutions are obtained and discussed for mass numbers ranging from hydrogen to the heavy gases. Characteristic relaxation times for approach to the steady state are estimated and found generally to be a small fraction of the synodic period. It is concluded that in all cases a marked anisotropy of molecular fluxes can be expected. By measuring these fluxes conclusions can be drawn about the distribution of gas sources, the physical properties of the surface and the composition of the lunar atmosphere.