Hydrogen diffusion in monoclinic and tetragonal zirconium oxides has been studied by electronic state calculations. In both structures, the optimized hydrogen site lies near the center of a distorted fluorite structure. The activation energy was calculated to be 120-200 kJ/mol, which is similar to experimentally measured values. The effects of compressive stress, alloying elements, and oxygen defects are considered individually. Compressive stress reduces the hydrogen diffusion coefficient by 40%/GPa. Oxygen defects and substituted Fe and Cr are thought to act as trapping sites for hydrogen, which probably reduces hydrogen diffusion in zirconium oxide.