Electron scattering by oxygen monolayers on the Ru(0 0 0 1) surface is studied both experimentally and theoretically. Sharp transmission resonances at low energies are revealed and established to originate from critical points of a special kind in the complex band structure of the substrate. Electron reflection from the clean and oxidized Ru(0 0 0 1) is measured for kinetic energies up to 40 eV at normal incidence for oxygen coverages of 1/4, 1/2, 3/4, and one monolayer. The reflection spectra R(E) are analyzed using a Bloch-waves based ab initio scattering theory. In addition to the substrate-induced resonances the reconstructed (2 × 1) and (2 × 2) surfaces show surface resonances due to pre-emergent secondary diffraction beams. The R(E) spectra are shown to give unambiguous evidence of the hcp stacking of the oxygen layer.