Diffusion of a single Cu adatom on low-index Cu surfaces with different morphologies (with and without the presence of other Cu adatoms as well as near and over stepped surfaces) is studied using the embedded-atom method and a molecular static simulation. Migration energies of a Cu adatom in the presence of other Cu adatoms which are relevant in computer simulations of island growth are calculated. We have also calculated the formation and migration energies of an adatom and a vacancy in different layers as well as formation energies of steps on various Cu surfaces. Step-step interaction is shown to be repulsive and consistent with elasticity theory. Our calculations predict a lower activation energy for diffusion of a vacancy than of an adatom for all three Cu surfaces.