I. D. Mayergoyz (1986) has shown that the necessary and sufficient conditions for a hysteretic process can be described by a classical Preisach model are that it possesses the wiping-out property and the congruency property. Magnetization-dependent Preisach models also have the wiping-out property but do not possess the congruency property. The necessary and sufficient conditions for a process to be describable by either of two magnetization-dependent models, the product model and the moving model are presented. The identification problem for both of these models is also addressed. Both models require the wiping-out property, but the moving model requires the skew-congruency property, and the product model requires nonlinear congruency.