A general geometrical transformation of the coordinates and of the displacement field is proposed; it is used to convert any boundary value problem for a linear elastic body into another one with different geometry, elastic moduli and boundary conditions. With this method, new problems, especially for inhomogeneous anisotropic bodies, may be solved by use of solutions of simpler ones. After a derivation of sufficient conditions to be fulfilled by such a transformation, the case of a linear homogeneous transformation is investigated in more detail. It is shown that a number of situations exist for which the transformed problem has a known analytical solution which can be used to derive the solution of the original problem straightforwardly. Special attention is paid to Saint-Venant-type anisotropy and to the derivation of the Green function for an infinite or a semi-infinite body.