The term elasticity seems to conjure different images in different minds. After a discussion of the various interpretations of elasticity espoused by the pioneers, we discuss the notions of Cauchy elastic and Green elastic bodies, and whether Cauchy elastic bodies that are not Green elastic are reasonable from a physical standpoint. We then discuss a class of models, more general than classical Cauchy elastic bodies, and we find that such bodies need not be Green elastic. While a stored energy can be associated with these materials, the stress is not derivable from the stored energy. One can delineate conditions under which these models are thermodynamically consistent in that they meet the second law of thermodynamics; more precisely, the general class of bodies that is being described is incapable of dissipation1 in any process whatsoever. These models not only add to the repertoire of the elasticians in modeling solids that are incapable of dissipation, but also they seem to provide an opportunity for a genuinely new approach to the study of problems that result in singularities within the classical theory of linearized elasticity, such as that encountered in the rupturing and fracturing of solids. The generalized framework also provides a rational basis for developing linearized theories within which the linearized strain bears a non-linear relationship to the stress.