One goal of this paper is to discuss the classical definition of granular temperature as an extension of its thermodynamic equivalent and a useful concept which provides an important characterization of fluidized granular matter. Following a review of some basic concepts and techniques (with emphasis on fundamental issues) we present new results for a system that can exhibit strong violations of equipartition, yet is amenable to description by classical granular hydrodynamics, namely a binary granular gas mixture. A second goal of this article is to present a result that pertains to dense granular and molecular solids alike, namely the existence of a correction to the elastic energy which is related to the heat flux in the equations of continuum mechanics. The latter is of the same (second) order in the strain as the elastic energy. Although recent definitions of temperatures for granular matter, glasses and other disordered many-body systems are not within the scope of this article we do make several general comments on this subject in the closing section.
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