We have used our recently developed self-consistent film-linearized muffin-tin orbital (FLMTO) technique to calculate the band structure and corresponding layer projected density of states for systems consisting of a monolayer of copper on top of a transition metal substrate. The two systems which have been considered (using five-layer slab geometries) and compared to pertinent experiments are (111)Cu on (0001)Ru and (100)Cu on (100)Ni. For the Cu monolayer on Ru we obtain close agreement with the experimentally observed (via UPS) downward shift (compared to bulk copper behavior) of the top of the copper d bands relative to the Fermi energy. Also there is charge transfer from the interface ruthenium to the surface copper. For Cu/Ni, the theoretical behavior reproduces the experimentally observed (via EELS) filling and narrowing of the surface d bands.