The main subject of this review is the consideration of catalytic oxidation reactions, which are greatly influenced by solid state effects in the catalyst material. Emphasis is laid upon the correlation between the presence of mobile ionic defects, together with the associated ionic conductivity, and the catalytic performance. Both total and selective oxidation reactions and oxidative conversion reactions are considered. Well-known examples of such behaviour include oxidative methane conversion with lanthanide oxides, carbon monoxide oxidation on fluorite type catalysts, selective olefin oxidation using vanadia based catalysts, etc. Furthermore, because oxygen exchange between gas and solid is always part of the oxidation process, this is considered too. The discussion of the application of the solid oxides under consideration to practically important oxidation processes, together with the influence thereon of their solid state properties, forms a major part of this review. Computational modelling and simulation of catalyst structure and behaviour is also considered. Special attention is given to the potentialities offered by using ionic and mixed conducting oxides either as the electrode material in a solid electrolyte fuel cell (SOFC) or as a separating, dense membrane in a ceramic membrane reactor. The use of porous membranes in such reactors is also taken into consideration. On the one hand these may be used to study the above relationship between catalytic behaviour and solid state properties, on the other hand to obtain a reactor configuration allowing better use of reactants and/or catalysts. Besides the controlled supply of (or removal) of oxygen to (or from) the side where the catalyst and the reactants are located, a promising feature of both experimental approaches is that the oxygen flux may alter the relative presence of different oxygen species (O-2, O, O-2(-). O-2(2-), O-3(-), O-, etc.) on the catalyst surface. In this way species are provided having a strong influence on the selectivity for partial oxidation reactions and oxidative conversion reactions.