Oxygen vacancy migration is studied in monoclinic HfO2 and across its interface with SiO2 through density functional calculations. In HfO2, long-range diffusion shows activation barriers of 2.4 and 0.7 eV for the neutral and doubly positively charged vacancy, respectively. In the latter case, the migration preferentially occurs along one-dimensional pathways. A HfO2/SiO2 interface model is constructed to address O vacancy migration across high-kappa gate stacks. The vacancy is shown to stabilize in its neutral charge state upon entering the SiO2 layer.