In order to model correctly the corrosion of spent nuclear fuel under disposal conditions, it is important to understand its behavior in the presence of oxidants. To advance in this direction, we consider the oxidation of UO2. We investigate computationally the adsorption of various species on its three most stable surfaces: (111), (110), and (100), with emphasis on incorporating a full non-collinear PBE+U approach. Various species, namely O, O2, H2O and H2O2 are considered due to their relevance for the oxidation of UO2. The disso- ciation energy and an estimate for the dissociation barrier for O2 were obtained, using the preferred adsorption configurations of O and O2. The adsorption configurations for H2O in our study compare well with previous studies that used collinear approximations, both in terms of relative stability of configurations and bond lengths. Differences in adsorption energies were found, which may be important for reaction kinetics. Dissociative reactions in which the water molecule splits in hydrogen and hydroxyl occur only on one of the three surfaces. The hydrogen further reacts with a surface oxygen to also form a hydroxyl group. Not surprisingly, we find that H2O2 binds more strongly to the three surfaces than water (lower formation energy), and similar to H2O adsorption, dissociative reactions may occur. The dissociated hydrogen reacts with a surface oxygen to form a hydroxyl group and the hydroperoxyl molecule binds with a surface uranium. Our study, which includes a detailed study of electron transfer, magnetic structure and the preferred adsorption configurations, gives insight into the uranium oxidation states and the influence of surface geometry on adsorption. The findings contribute to a more comprehensive understanding of the early stages of UO2 oxidation.