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This thesis is concerned with the development of boundary element techniques for the numerical solution of several important classes of axisymmetric heat conduction and thermoelastic problems involving nonhomogeneous solids with material properties that vary continuously in space. The problems under consideration have applications in the analyses of functionally graded materials which play an important role in engineering. The classes of axisymmetric problems considered in this thesis may be categorized as follows: (a) nonsteady heat conduction in a nonhomogeneous solid with temperature dependent material properties, (b) nonclassical heat conduction, based on the dual-phase-lag heat conduction model, in a nonhomogeneous solid, and (c) thermoelastostatic and thermoelastodynamic deformations in nonhomogeneous solids. to complete my studies successfully.