The process of deformation is interpreted as a process of continuing the solution to the parameter, as which can be set to load or also true shift in the static problems, time in dynamic problems, or the best continuation parameter in those and other cases. This avoids several problems in traditional algorithms using Lagrangian coordinates and associated with substantial distortions of difference or finite-element meshes up to their degeneration. We obtain the general equations of the continuation using Euler coordinates to problems of nonlinear deformation of solids and structures, including those with complex rheology. Using the Euler coordinates allows us to construct solutions of these equations using simple Cartesian or polar grids.