Rapid solidification can produce materials with superior, and sometimes astonishing, physical and mechanical properties. This has led to many advanced materials processing techniques such as melt atomization, thermal spray coatings, melt-spinning, laser melting and resolidification, and high-energy beam treatment of surfaces, to name a few. Because of large melt undercooling and very high rate of solidification, the rapid solidification processes inherently involve nonequilibrium kinetics of phase change and coupled transport of energy and species. This coupling ultimately determines the phase selection, microstructure formation and final properties of the rapidly solidified materials. The fundamentals of rapid solidification, and recent progress made in modeling of these processes are reviewed in this article. Both single component and alloy materials are considered, and the issues related to thermodynamics of rapid solidification, nucleation and growth kinetics of crystalline phase, planar interface stability, dendritic growth theory, and microsegregation are discussed. Thermal transport models incorporating the basic principles of rapid solidification and their integration with kinetics models are presented. Special features of theoretical results are discussed in detail, particularly, for phase selection and microstructure formation in atomized powders, pulsed laser processing, and melt-substrate quenching.