The analytical studies on contact mechanics have been limited to problems of either half space or single layer due to mathematical difficulty, though layered composites subjected to indentation are commonly encountered in industrial applications. The studies of indentation of layered composites, however, are based on numerical approaches. This dissertation provides a theoretical method for the contact mechanics of layered composites subjected to axially symmetric indentation. A new function is introduced in this dissertation to reduce the complexity of the mathematical process, and mathematical solutions are provided for all the problems investigated in this research. However, the mathematical solution for the final integration could only be obtained for the point loading condition. A numerical method was used to evaluate the final results for other loading conditions, such as uniform stress, flat indentation and spherical indentation. In this dissertation, the effects of material property, layer thickness, boundary condition, loading condition, and lamination on contact mechanics were investigated for the cases of a half space, a single layer bonded to a rigid base, a single layer bonded to an elastic half space and two-layered composites bonded to a rigid base. The dissertation also investigated the frictional effect at the contact interface. Both shear slip and normal separation theories were incorporated into the mathematical formulation, allowing the study of debonding at the interface between layers. New transformed shear slip and normal separation coefficients are proposed to study the imperfect bonding interfaces with a finite length. Contact mechanics models have been proposed based on numerical results. These models provide insight into the relationships among total load, maximum displacement, contact radius, layer thickness and material properties, and guidelines for engineering applications.