The principles of forming metastable solids by rapid melt quenching are reviewed with primary emphasis on the growth of crystals into melts and glasses. Experience is reviewed which indicates that the rate of growth of crystals in pure metal melts may be limited only by the impingement frequency of atoms from the melt on to the crystal face. However, growth requiring impurity distribution will be governed by the more highly correlated atomic transport processes within the melt. We have suggested that the shapr increase in the resistance to this redistribution with falling temperature is an important requirement for melt quenching alloys to the glassy state. Growth of crystals into covalently bound melts and glasses is, in contrast with that in pure metal melts, limited by thermal activation barriers with magnitudes near those of the covalent bond energies. The formation of ledges on densely packed crystal faces and the role of such ledges in crystal growth are discussed. The experience on the regrowth of crystalline silicon into amorphous overlays is discussed in terms of the concepts presented.