  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.
