We investigate the atomic and electronic structure of ultra-thin ZnO films (1 to 4 layers) on the (111) surfaces of Ag, Cu, Pd, Pt, Ni, and Rh by means of density-functional theory. The ZnO monolayer is found to adopt an α-BN structure on the metal substrates with coincidence structures in good agreement with experiment. Thicker ZnO layers change into a wurtzite structure. The films exhibit a strong corrugation, which can be smoothed by hydrogen (H) adsorption. An H over-layer with 50% coverage is formed at chemical potentials that range from low to ultra-high vacuum H2 pressures. For the Ag substrate, both α-BN and wurtzite ZnO films are accessible in this pressure range, while for Cu, Pd, Pt, Rh, and Ni wurtzite films are favored. The surface structure and the density of states of these H passivated ZnO thin films agree well with those of the bulk ZnO(0001¯)-2×1-H surface.