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Several common modes of crystal growth provide particularly simple and elegant examples of spontaneous pattern formation in nature. Phenomena of interest here are those in which an advancing nonfaceted solidification front suffers an instability and subsequently reorganizes itself into a more complex mode of behavior. The purpose of this essay is to examine several such situations and, in doing this, to identify a few new theoretical ideas and a larger number of outstanding problems. The systems studied are those in which solidification is controlled entirely by a single diffusion process, either the flow of latent heat away from a moving interface or the analogous redistribution of chemical constituents. Convective effects are ignored, as are most effects of crystalline anisotropy. The linear theory of the Mullins-Sekerka instability is reviewed for simple planar and spherical cases and also for a special model of directional solidification. These techniques are then extended to the case of a freely growing dendrite, and it is shown how this analysis leads to an understanding of sidebranching and tip-splitting instabilities. A marginal-stability hypothesis is introduced; and it is argued that this intrinsically nonlinear theory, if valid, permits aone to use results of linear-stability analysis to predict dendritic growth rates. The review concludes with a discussion of nonlinear effects in directional solidication. The nonplanar, cellular interfaces which emerge in this situation have much in common with convection patterns in hydrodynamics. The cellular stability problem is discussed briefly, and some preliminary attempts to do calculations in the strongly nonlinear regime are summarized.