Abstract：A concise mathematical theory is summarized which was developed by the author over the last twenty years for analytically deriving solutions of three-dimensional boundary value problems encountered in elastic materials whose properties vary，continuously or in discrete steps，with depth from a surface exactly within the framework of elasticity. Such materials are now named as functionally graded materials(FGM). Some results are also presented that the author has obtained by using the theory in the analysis of various engineering problems. Such problems include elastodynamics，thermoelasticity，effect of imperfect bonding，ground land subsidence due to coal mining，design and evaluation of pavement structures，ground investigation with static cone penetration，soil consolidation，as well as fracture mechanics in functionally graded materials.
