Abstract:Numerical study of alloy solidification processes has seen great developments since the 1980's. The major mathematical models applied to macroscopic processes are discussed. The basic ideas, numerical methods, relative merits and disadvantages of those models are presented. This paper indicates that microscopic processes have a very important influence on the macroscopic transport phoenomena and the two aspects must be considered at the same time while building themathematical model. The relevant engineering applications and future directions are also discussed.