The large interest in the properties of transition metal surfaces has been recently fostered by inelastic helium atom scattering experiments carried out by Hulpke and Luedecke. Sharp and giant anomalies in the surface phonon dispersion curves along Gamma-H and Gamma-S have been detected on W(110) and Mo(110) at a coverage of one monolayer of hydrogen. At the same critical wave vectors a smaller second indentation is present in the experimental phonon branches. Recently we have proposed a possible interpretation which is able to explain this and other experiments. In this paper we discuss results of our recent ab initio calculations of the atomic and electronic properties of the clean and H-covered Mo(110) surface in more details. For the full monolayer coverage the calculated Fermi-surface contours are characterized by strong nesting features which originate the observed anomalies.