This chapter deals with formulations based on boundary integral equations (BIEs) for elastic and plastic problems. After a brief review of the basic integral identities of solid mechanics and issues associated with the singular character of the fundamental solutions, the collocation and symmetric Galerkin BIE formulations and the associated boundary element methods (BEMs) are presented in their conventional form where the complete matrix equation is set up using numerical integration and stored. This approach being inadequate for large problem sizes, fast solution techniques are then reviewed, with emphasis on the fast multipole method. Next, BEMs in both collocation and symmetric Galerkin form are described for fracture mechanics and small-strain elastoplasticity. The treatment of the hypersingular integrals arising in the integral representation of tractions on the crack surface or of strains at potentially plastic interior points is discussed, along with other issues. Shape sensitivity analysis techniques are also presented, based on either the direct differentiation of the primary elastic BIE or an adjoint solution. Finally, the symmetric Galerkin BIE is used to define a symmetric formulation for BEM–FEM coupling.