We investigate at the atomic scale the oxygen diffusion process occurring during silicon oxidation. First, we address the energetics of several oxygen species in the oxide using density-functional calculations. Our results support the interstitial O-2 molecule as the most stable oxygen species. We then adopt a classical scheme for describing the energetical and topological properties of the percolative diffusion of the O-2 molecule through the interstitial network of the oxide. By studying a large set of disordered oxide structures, we derive distributions of energy minima, transition barriers, and the number of connections between nearest-neighbor minima. These distributions are then mapped onto a lattice model to study the long-range O-2 diffusion process by Monte-Carlo simulations. The resulting activation energy for diffusion is found to be in agreement with experimental values. We also extend our atomic-scale approach to an oxide of higher density, finding a significant decrease of the diffusivity. To address the O-2 diffusion directly at the Si-SiO2 interface, we construct a lattice model of the interface which incorporates the appropriate energetic and connectivity properties in a statistical way. In particular, this lattice model shows a thin oxide layer of higher density at the interface, in accord with x-ray reflectivity data. We carry out Monte-Carlo simulations of the O-2 diffusion for this model and obtain the dependence of the diffusion rate on oxide thickness. For oxide thicknesses down to about 2 nm, we find that the presence of an oxide layer of higher density at the Si-SiO2 interface causes a drop of the O-2 diffusion rate with respect to its value in bulk SiO2, in qualitative agreement with the observed oxidation kinetics.