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In this paper we review two important theoretical areas to which J. W. Cahn has made major contributions: (i) The theory of the ξ-vector developed by Hoffman and Cahn, which provides an elegant setting for the description of the equilibrium shapes of sharp interfaces in the presence of anisotropic surface energy. (ii) Diffuse interface theories of phase transitions. We describe recent work which connects these two complementary facets of models of interfaces by the development of a generalized ξ-vector for diffuse interface models with anisotropic surface energy. We show that the generalized ξ-vector plays a central role in both the mathematical and physical aspects of a wide range of diffuse interface theories of interfaces with either anisotropic surface energy or attachment kinetics.