We report extensive density functional theory calculations, using pseudopotentials with a plane-wave basis, for the properties of the (010) face of molybdenum trioxide (alpha-MoO3). The surface is modeled by a one-layer slab. Calculated bond Lengths compare favorably with experimental measurements. The bonding of the different oxygen species to molybdenum is analyzed using the crystal orbital overlap population. This analysis indicates that the bonding is a combination of ionic and covalent character for all oxygen species. The terminal oxygen exhibits covalent bonding to Mo which is stronger than either of the two bridging oxygens. We also study the adsorption of hydrogen on this surface. Hydrogen is most strongly adsorbed over the terminal oxygen, followed by the asymmetric bridging oxygen, and then the symmetric bridging oxygen. This trend is explained in terms of simple chemical concepts. The inclusion of full surface relaxation is important for even a qualitative description of adsorbate bonding.