Elastic interactions arising from a difference of lattice spacing between two coherent phases can have a strong influence on the phase separation (coarsening) of alloys. If the elastic moduli are different in the two phases, the elastic interactions may accelerate, slow down or even stop the phase separation process. If * Dedicated to John W. Cahn, on the occasion of his 70th birthday 1 the material is elastically anisotropic, the precipitates can be shaped like plates or needles instead of spheres and can form regular precipitate superlattices.Tensions or compressions applied externally to the specimen may have a strong effect on the shapes and arrangement of the precipitates. In this paper, we review the main theoretical approaches that have been used to model these effects and we relate them to experimental observations. The theoretical approaches considered are (i) 'macroscopic' models treating the two phases as elastic media separated by a sharp interface (ii) 'mesoscopic' models in which the concentration varies continuously across the interface (iii) 'microscopic' models which use the positions of individual atoms.