The partial and total oxidation of methanol on clean and oxygen-precovered Cu(110) has been studied by periodic density functional theory calculations within the generalized gradient approximation. Reaction paths including the geometry and the energetics of several reaction intermediates and the activation barriers between them have been determined, thus creating a complete scheme for methanol oxidation on copper. The calculations demonstrate that the specific structure of oxygen on copper plays an important role in both the partial and the total oxidation of methanol. For lower oxygen concentrations on the surface, the partial oxidation of methanol to formaldehyde is promoted by the presence of oxygen on the surface through the removal of hydrogen in the form of water, which prevents the recombinative desorption of methanol. At larger oxygen concentrations, the presence of isolated oxygen atoms reduces the C-H bond breaking barrier of adsorbed methoxy considerably, thus accelerating the formation of formaldehyde. Furthermore, oxygen also promotes the formation of dioxymethylene from formaldehyde, which then easily decays to formate. Formate is the most stable reaction intermediate in the total oxidation. Thus the formate decomposition represents the rate-limiting step in the total oxidation of methanol on copper.