The thermodynamic, structural and electronic properties of Cu-CeO(2) (ceria) surfaces and interfaces are investigated by means of density functional theory (DFT+U) calculations. We focus on model systems consisting of Cu atoms (i) supported by stoichiometric and reduced CeO(2) (111) surfaces, (ii) dispersed as substitutional solid solution at the same surface, as well as on (iii) the extended Cu(111)/CeO(2)(111) interface. Extensive charge reorganization at the metal-oxide contact is predicted for ceria-supported Cu adatoms and nanoparticles, leading to Cu oxidation, ceria reduction, and interfacial Ce(3+) ions. The calculated thermodynamics predict that Cu adatoms on stoichiometric surfaces are more stable than on O vacancies of reduced surfaces at all temperatures and pressures relevant for catalytic applications, even in extremely reducing chemical environments. This suggests that supported Cu nanoparticles do not nucleate at surface O vacancies of the oxide, at variance with many other metal/ceria systems. In oxidizing conditions, the solid solutions are shown to be more stable than the supported systems. Substitutional Cu ions form characteristic CuO(4) units. These promote an easy and reversible O release without the reduction of Ce ions. The study of the extended CeO(2)(111)/Cu(111) interface predicts the full reduction of the interfacial ceria trilayer. Cu nanoparticles supported by ceria are proposed to lie above a subsurface layer of Ce(3+) ions that extends up to the perimeter of the metal-oxide interface.