Oxygen and hydrogen mobility are among the important characteristics for the operation of solid oxide fuel cells, permselective membranes and many other electrochemical devices. This, along with other characteristics, enables a high-power density in solid oxide fuel cells due to reducing the electrolyte resistance and enabling the electrode processes to not be limited by the electrode-electrolyte-gas phase triple-phase boundary, as well as providing high oxygen or hydrogen permeation fluxes for membranes due to a high ambipolar conductivity. This work focuses on the oxygen and hydrogen diffusion of mixed ionic (oxide ionic or/and protonic)–electronic conducting materials for these devices, and its role in their performance. The main laws of bulk diffusion and surface exchange are highlighted. Isotope exchange techniques allow us to study these processes in detail. Ionic transport properties of conventional and state-of-the-art materials including perovskites, Ruddlesden–Popper phases, fluorites, pyrochlores, composites, etc., are reviewed.