A comprehensive description of the science and applications of the undercooling phenomenon is given. It is composed of several main parts: experimental techniques for undercooling; characterization of the undercooled melt as the first step in rapid solidification; introducing the concepts of modern theories of rapid dendrite and eutectic growth and their comparison with experimental results, and a survey of metastable materials formed from the non-equilibrium state of an undercooled melt.