Uranium dioxide (UO2) is the main component of nuclear fuels. This compound is very susceptible to oxidation, which results in the formation of the mixed-valence oxide U3O8. [1] From an application point of view, this interest relates to the safe and sustainable management of nuclear fuel. Our goal is an understanding of this oxidation process at the atomic level, using ab initio electronic structure calculations. The focus of this work lies in the early oxidation of stoichiometric UO2, where excess oxygen can be treated as interstitial point defects. In order to study this oxidation we investigate the adsorption of the O atom and oxygen-containing molecules (O2, H2O, H2O2) on the three most stable UO2 surfaces ((111), (110), and (100)). The calculations are performed within DFT, using PBE+U including spin-orbit coupling and taking into account the non-collinear 3k (transverse) magnetic ground state. Dissociative adsorption is found for H2O and H2O2 on the (111) and (100) UO2 surfaces respectively, forming hydroxyl groups on the surface. Adsorption on the UO2 surface lowers the barrier for the dissociation of O2. The defect formation energy is calculated for interstitial oxygen, up to several layers into the material starting from the surface. Additionally, adsorption energies are studied as a function of temperature and partial pressure.