The problem under consideration is the scattering of elastic waves by inhomogeneous obstacles. The main goal is to obtain approximation techniques which are amenable to numerical implementation. For time-periodic problems a coupling procedure involving finite elements and boundary integral equations is described. For general time-dependent problems, artificial boundary methods are studied. In both cases the concept of generalized stress, as originated by Kupradze, plays a central role. The analysis is restricted to planar two-dimensional problems since these illustrate the essential ideas.