This paper is a modern synthesis of the more than seventy years of literature on the elasticity of substitution. At times, authors, such as Mundlak, have provided syntheses of the literature but these often seem to be forgotten by later authors and particularly by applied economists. I synthesize several of these approaches with the full development of duality theory. There are many different legitimate definitions of the ES and the elasticity of complementarity (EC). None of these is the one true ES - which one is useful depends on what we wish to measure. As their value and even sign can vary dramatically, the choice of the appropriate indicator is important. I propose a classification scheme: primal vs. dual measures, gross vs. net measures, ratio, scalar and mixed elasticities, and four different basic concepts of substitution and complementarity. Additionally, the Pigou Elasticity of Complementarity is reintroduced after seventy years of obscurity and a new derivation of the basic Hicks (or direct) ES is provided. The implications for the energy-capital controversy are also briefly discussed and an empirical example shows how the various elasticities differ.
