I will describe the characterization of heat transfer in models of high-dimensional billiards, which capture the properties of materials that possess both the spatial structure of solids and collisional dynamics of gases. I will discuss the conditions under which the heat conductivity of these models can be universally expressed in terms of the frequency of collisions between gas particles. This universality manifests itself when interactions between gas particles are rare, though not necessarily small. The dynamical properties of these models will be analyzed, and the role of dimensionality emphasized.