AbstractWe offer a revised exposition of the three types of heat-propagation theories proposed by Green and Naghdi. Those theories, which make use of the notion of thermal displacement and allow for heat waves, are at variance with the standard Fourier theory; they have attracted considerable interest, and have been applied in a number of disparate physical circumstances, where heat propagation is coupled with elasticity, viscous flows, etc. (Straughan in Heat waves. Applied mathematical sciences, vol. 177. Springer, Berlin, 2011). However, their derivation is not exempt from criticisms, that we here detail, in hopes of opening the way to reconsideration of old applications and proposition of new ones.