In the absence of surface tension and external force fields, the equilibrium between a hydrostatically stressed crystal and its melt is neutral with respect to the perturbations associated with particle transfer from one region of the boundary into another. However, under the action of arbitrary small nonhydrostatic components of the stress field in the elastic crystal, the neutral equilibrium is transformed to an unstable equilibrium . This instability is very general in nature; for example, for it to be seen the liquid media need only to be able to dissolve the solid phase or in some way to assist the transport of particles along the crystal's surface. In contrast, the surface tension, roughly speaking, stabilizes the shape of the interphase boundary but it cannot suppress the instability generated by the nonhydrostatic components of the stress field in the region of sufficiently long perturbations. Until now the basic instability mechanism discussed here seems to have escaped the attention of theorists. This mechanism allows one to look in a completely new way at a broad range of phenomena. We discuss tentative manifestations and role of this instability in low temperature physics, in materials science, in theory of crystal growth, and, in particular, in theory of epitaxy and of the Stranski-Krastanow pattern of growth of thin crystalline films.