We have investigated the formation of native defects in anatase TiO2 using density functional theory (DFT) modified with on-site Coulomb terms (DFT+U) applied to both Ti d and O p states. Oxygen vacancies and titanium interstitials are found to be deep donors that trap two and four electrons, with transition levels that explain the two features seen in deep level transient spectroscopy experiments. Titanium vacancies are deep acceptors accommodating four holes. Self-trapping of both electrons and holes is also predicted. In all cases both donor and acceptor trap states correspond to strongly localized small polarons, in agreement with experimental EPR data. Variation in defect formation energies with stoichiometry explains the poor hole-trapping of reduced TiO2.