One of the current and widely used non-destructive testing methods for monitoring and determining the elastic properties of materials is nanoindentation. In this case, to interpret the test results, a non-trivial task arises of constructing an adequate mathematical model of the indentation process. As a rule, in many cases, analytical formulas are used that are obtained from an elastic linear formulation of problems about the introduction of a non-deformable stamp into a homogeneous elastic half-space. Currently, the numerical formulation of the problem makes it possible to obtain and use a numerical solution obtained taking into account the complete plastic nonlinear behavior of the material. In this work, a study of contact problems on the introduction of a spherical and conical indenter into an elastoplastic homogeneous half-space was carried out. To verify the numerical solution, the problem of introducing a spherical and conical indenter into an elastic homogeneous half-space was also solved and compared with known analytical solutions. Issues of convergence and tuning of numerical methods, the influence of plasticity and the applicability of analytical solutions are explored. Problems are solved numerically using the finite element method in the Ansys Mechanical software package.