A description of the method for solving various electrostatic problems associated with macroscopic bodies of arbitrary shape is given. The desired potential is sought in the form of a system decomposition of eigenfunctions, which in turn are regular solutions of the Laplace equation in the absence of an external electric field. This method found common expressions for the dipole polarizability tensor of the body, for the Green function, as well as for the potentials of the edge (boundary) Dirichlet and Neumann problems. A number of precisely solvable examples (ball, spheroids, wedge, cone, etc.) are considered, for each of which a complete system of intrinsic functions is found. The above approach can be used for other physical and mathematical problems that require solving the Laplace equation.