Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.