This is the first of two papers in which we give a systematic account of firstand second-order moduli of elasticity (or elasticities) for finitely deformed elastic materials, particular emphasis being placed on the discussion of second-order moduli. In the present paper we are concerned with general elastic materials. The results are then particularised to isotropic materials in the paper which follows and the inherent simplicity of this special case exploited with the aid of a powerful theorem of tensor calculus. We confine attention throughout to compressible materials but the extension to incompressible materials is straightforward. A connected treatment of first-order elasticities for finitely strained elastic materials has been presented by TRUESDELL & NOLL [1965, w on the basis of formulations given by TRUESDELL [1961] and by TOUPIN & BERNSTEIN [1961, w The approach of these authors is first to introduce two sets of first-order elasticities, each defined as components of the fourth-order tensor obtained by differentiating a stress tensor with respect to a measure of deformation, and then to construct from them further first-order moduli. Lacking in the resulting theory are considerations governing the initial choice of stress-deformation pairs, transparent interpretations of the derived moduli, and a natural extension of the definition procedure to higher-order moduli. We believe that each of these requirements is met in the course of the discussion begun in this section and elaborated in Section 2. In the applications of elasticity theory which have so far been studied without recourse to approximation, second-order moduli arise most naturally in connexion with the variation in strength of acceleration waves propagating in deformed elastic materials. This aspect of elastic wave theory was first studied by W. A. GREEN [1964, 1965] in the special context of isotropic materials and subsequent contributions have mainly accepted the same restriction (see CrmN [1968, 1, 2, 3], BLAND [1969, w JUNEJA • NARIBOLI [1970]). It is evident, however, from the work of VARLEY [1965, w on the propagation of acceleration waves in viscoelastic materials*, that the analysis carries over without essential change to general elastic materials and this generalisation has in part been realised in recent