In this study the contact problems of thin films bonded to Functionally Graded Materials (FGM) are considered. In these problems the loading consists of any one or combination of stresses caused by uniform temperature changes and temperature excursions, far field mechanical loading, and residual stresses resulting from film processing or in the manufacturing process of the graded coating. The primary interest in this study is in examining stress concentrations or singularities near the film ends for the purpose of addressing the question of crack initiation and propagation in the substrate or along the interface. The underlying contact mechanics problem is formulated by assuming that the film is a "membrane" and the FGM an elastic continuum, and is solved analytically by reducing it to an integral equation. The calculated results are the interfacial shear stress between the film and the graded substrate, mode II stress intensity factor at the end of the film and the axial normal stress in the film.