There is no air on the Moon, but it has recently been announced that water is definitely present. In 2013 NASA funded Lunar research found evidence of water locked in mineral grains, deep beneath the surface.
Using data from NASA’s Moon Mineralogy Mapper (M3) instrument aboard the Indian Space Research Organization’s Chandrayaan-1 spacecraft, scientists remotely detected magmatic water
Image of the central peak of Bullialdus crater, a Constellation region of interest. LROC Narrow Angle Camera image M114098458LE, image is 500 m wide Credit: NASA/GSFC/Arizona State University