Thermoelasticity theories predicting a finite speed for the propagation of thermal signals have come into existence during the past 20 years. In contrast to the conventional thermoelasticity theory, these nonclassical theories involve a hyperbolic-type heat transport equation, and are motivated by experiments exhibiting the actual occurrence of wave-type heat transport (second sound). Several authors have formulated these theories on different grounds, and a wide variety of problems revealing characteristic features of the theories has been investigated. This article presents a fairly self-contained bibliographical review of the relevant literature. Novelties involved in the formulations of the theories are emphasized, and concise derivations of the governing equations presented. Results concerned with solutions of initial-boundary value problems are summarized, and salient aspects of the theories illustrated. The list of references is exhaustive and up-to-date.