The migration of ions in HfOx was investigated by means of large-scale, classical molecular-dynamics simulations over the temperature range 1000≤T/K≤2000. Amorphous HfOx was studied in both stoichiometric and oxygen-deficient forms (i.e., with x = 2 and x = 1.9875); oxygen-deficient cubic and monoclinic phases were also studied. The mean square displacement of oxygen ions was found to evolve linearly as a function of time for the crystalline phases, as expected, but displayed significant negative deviations from linear behavior for the amorphous phases, that is, the behavior was sub-diffusive. That oxygen-ion migration was observed for the stoichiometric amorphous phase argues strongly against applying the traditional model of vacancy-mediated migration in crystals to amorphous HfO2. In addition, cation migration, whilst not observed for the crystalline phases (as no cation defects were present), was observed for both amorphous phases. In order to obtain activation enthalpies of migration, the residence times of the migrating ions were analyzed. The analysis reveals four activation enthalpies for the two amorphous phases: 0.29 eV, 0.46 eV, and 0.66 eV (values very close to those obtained for the monoclinic structure) plus a higher enthalpy of at least 0.85 eV. In comparison, the cubic phase is characterized by a single value of 0.43 eV. Simple kinetic Monte Carlo simulations suggest that the sub-diffusive behavior arises from nanoscale confinement of the migrating ions.