A meshless continuum formulation is presented. The formulation resembles the element free Galerkin method, where discrete nodes (or particles) are mutually attached via shape functions constructed from moving least squares approximations. No reference to any type of mesh is made. The performance of the present formulation is superior to the performance of other methods in problems involving strain localization, fracture, contact, or extreme deformations. This paper documents the construction of discrete equations as well as continuum-mechanical details concerning the behavior of ductile as well as brittle solids. In the meshless analysis both Eulerian and Lagrangian shape functions can be applied. Eulerian shape functions are calculated from the nodes' spatial coordinates and are constantly updated. In contrast, the Lagrangian shape functions are calculated from the nodes' convected material coordinates. The Lagrangian shape functions provide the most stable simulations, whereas Eulerian shape functions lead to numerical instabilities. It is demonstrated how the meshless formulation provides an objective framework for studying crustal strain localizations as the meshless method does not suffer from the mesh sensitivity of the finite element method. In addition, examples on applications in fracture and contact problems are given through ductile necking and folding.