Electrical percolation in two-phase materials involves a very singular behavior, manifested as a huge change in the electrical conductivity, for a given volume or mass fraction of the phase with higher conductivity. In contrast, in the case of heat transfer, in two-phase composite systems, analogous percolative phenomena are far more elusive and have been rather difficult to observe in various physical systems. In this Perspective, we present a critical analysis of experimental results and the application of theoretical models aimed to study the effects of percolation phenomena on the thermal and electrical properties of two-phase materials. Our attention will be focused on composites made of high conductivity particles in a polymeric matrix. The effect of several factors, such as the geometrical and physical characteristics of fillers and their connectivity with the matrix, the proportion between the conductivity of filler and the matrix, as well as the crucial role of interfacial thermal resistance, is considered. In particular, the differences between the thermal and electrical thresholds and the physical and geometrical conditions that should be fulfilled to observe thermal percolation are discussed. Future trends, to be followed in the development of new materials, in order to enhance the thermal conductivity as well as in making the thermal percolative effects notable, based on including additional phases and 2D fillers, are also discussed.