The focus of this review is on the method of analytical development of thermoelastic problems for nonhomogeneous materials, such as functionally gradient materials (FGM). For such nonhomogeneous materials, both the thermal and mechanical material constants are described by the function of the variable of the coordinate system. Then, the governing equations for the temperature field and the associated thermoelastic field become of nonlinear form in general cases. Therefore, the theoretical treatment is very difficult and the exact solution for the temperature and the thermoelastic field is almost impossible to obtain. This nonlinear equation system is usually treated by introducing some linearization technique with appropriate theoretical approximation. In this review, the method of some analytical developments for the heat conduction problem and the associated thermal stress problem of a body with nonhomogeneous material properties is explained briefly, and some boundary value problems of technical interest, such as optimization problems for material nonhomogeneity and the problems of thermal stress intensity factor for a body with a crack, are discussed.