Abstract: Realizability toposes have been studied mainly in isolation, and not much attention has been devoted to morphisms between them. Also, it is not clear which toposes count as realizability toposes and which do not, so that we do not have a good idea what the category of realizability toposes would be. In this talk I will first present a working definition of what constitutes a realizability notion, and then present a category that explains how realizability toposes arise from combinatorial structures. A characterization of geometric morphisms between realizability toposes will be given, purely in terms of morphisms of the underlying combinatorial structures. This reduces the study of the 2-category of realizability toposes to a certain 2-category of combinatorial structures.