Elementary processes governing the epitaxial growth of metal oxides, like adsorption and diffusion, have been studied theoretically in the prototypical case of MgO/MgO(001). Most of these processes need to be investigated in an ab initio approach since they are usually accompanied by strong modifications of the electronic structure. This study has been carried out within the density functional theory, and complemented by molecular dynamics simulations using a rigid-ion potential. The growth of the oxide passes through several oxido-reduction reactions at the surface. The adsorption and the diffusion of Mg and O are strongly different from each other, and change considerably at steps from flat terraces. All these results point out the need of realistic models for the growth of metal oxides that take into account the chemical behaviour of the involved species as well as the influence of the defects.