The determination of the thermo-mechanical stress field in and around a spherical/cylindrical inhomogeneity surrounded by a functionally graded (FG) coating, which in turn is embedded in an infinite medium, is of interest. The present work, in the frame work of Boussinesq/Papkovich-Neuber displacement potentials method, discovers the potential functions by which not only the relevant boundary value problems (BVPs) in the literature, but also the more complex problem of the coated inhomogeneities with FG coating and sliding interfaces can be treated in a unified manner. The thermo-elastic fields pertinent to the inhomogeneities with multiple homogeneous coatings and various combinations of perfect/sliding interfaces can be computed exactly. Moreover, when the coatings are inhomogeneous, as long as the spatial variation of the thermo-elastic properties of the transition layer is describable by a piecewise continuous function with a finite number of jumps, an accurate solution can be obtained. The influence of interface conditions, stiffness of the core, spatial distributions of thermal expansion coefficient and shear modulus of FG coating, and loading condition on the stress field will be examined.