Changes in crystallographic structures can occur solely at low temperatures by a diffusionless, shear-type of transformation to a product structure termed martensite. This paper presents a review of martensitic transformations which occur in ferrous and nonferrous alloys. Specifically, kinetic, structure, and the theoretical approaches to describing diffusionless transformations are treated. The kinetic characteristics pertinent to this class of transformations include dependence upon temperature, time, applied stress, and chemical composition. The structural aspects of martensitic transformations include their morphology and crystallography. Four general types of martensite transformation morphology have been observed in steels and in nonferrous alloys. With particular reference to steels, the product structures may be morphologically described as either: (1) plates, (2) sheets comprised of martensite crystals or laths, (3) homogeneous sheets, or (4) surface martensite. Theories pertaining to martensitic transformations relate either to the initiation or to the subsequent growth of the product. Theories employing thermodynamic analyses or dislocation models generally are concerned with nucleation, while crystallographic theories attempt to predict the mode of growth and the final orientations of the product. Both approaches are discussed. Approximately 650 references appear in the extended bibliography which lists almost all articles published in English after 1940, in addition to major articles published in other languages. Tables which attempt to classify articles are also included to simplify research of martensite literature.