A brief coverage of the mechanics of contact problems is presented. The governing equations for three-dimensional axisymmetric elasticity problems in cylindrical coordinates are first formulated, which is followed by the solutions to classical problems of a concentrated force within an infinite medium (Kelvin problem), and a concentrated force at the boundary of a half-space (Boussinesq problem). The stress fields in a half-space loaded by an elliptical and a uniform pressure distribution over a circular portion of its boundary are presented. Indentation by a spherical ball and by a cylindrical circular indenter are analyzed. The second part of the chapter is devoted to Hertzian contact problems. The nonlinear force–displacement relation is derived for elastic contact of two spherical bodies pressed against each other by two opposite forces. The elastic contact of two circular cylinders is also considered. The contact pressure and the maximum shear stress are determined. The approach of the centers of the cylinders requires the consideration of the local contact stresses, as well as the stresses within the bulk of each cylinder.