The so-called superionic conductors represent a class of solid materials showing an unusually high ionic conductivity. Their structure is characterized by strong disorder in the sublattice of conducting ions. We shall describe the dynamic properties of the partially disordered state on the basis of two classes of theoretical models. For stochastic lattice gases we discuss the behavior of various time-correlation functions relevant for inelastic neutron- and light-scattering experiments and for tracer-diffusion measurements. In addition we study a system of interacting Brownian particles in the presence of a periodic potential as a model for optimized (AgI-type) ionic conductors. A mean-field theory is developed which implies a relationship between the conductivity and structural properties.