INTRODUCTIONResearch on the preparation and study of single-crystalline M 1 -N2+x (R = rare earth element, REE) materials with large deviations from stoichiometry began at the Institute of Crystallography of the Academy of Sciences of the USSR in 1963. The efforts were continued in cooperation with the chemistry staff of Moscow State University. Progress on the quest and preparation of crystals of nonstoichiometric fluorides for quantum electronics up to 1978 was reviewed in dissertations [2]. However, a large part of the data on growth of M 1 -N2+x single crystals is scattered over a large number of obscure publications.The purpose of the present article is to review and correlate all work published by us on the growth of M 1 -xRxf'2+x fluorite phase crystals.The uniqueness of this family of crystals and hence the unwavering interest in their preparation and study consists in the gross deviations from stoichiometry which permit fluorite structural types with heterovalent substitution of M2+ by R3+ (up to 50 mole % RF 3 ). Changes in the physical and physicochemical characteristics of the crystals are effected and controlled as a result of the gross nonstoichiometry. Mechanical, radiative, thermal, chemical, optical, transport, acoustic, and other structurally sensitive characteristics are affected.The M 1 -xRxf'2+x phases are the most striking with respect to nonstoichiometry and most feasible from the viewpoint of technological production among the more than 150 types of known fluorites with gross stoichiometric imbalance [1]. Numerically, they comprise a third of all known types (according to qualitative chemical compOSition) of fluorite-fluoride nonstoichiometric (two-component) phases.The scope of our program characteristically includes preparation of single crystalS of the M 1 -xRxf'2+x nonstoichiometric fluorite phases for the whole concentration region covering hundredths of a fraction of a percent to saturation, i.e., -50 mole % RF 3 . Numerous publications by other authors have appeared for REE trifluoride concentrations of 1-3 mole %. Review of these is a separate task. Few articles can be singled out on the preparation of single crystals with high contents of RF 3 . The most systematic of these should be mentioned [3][4][5][6]. In particular, the high content of heterovalent impurities in the CaF 2 structure is of prime 65 K. S. Bagdasarov et al. (eds.), Growth of Crystals