Abstract: We present some recent work motivated by the question of existence of traveling surface elastic waves of permanent form in the half-plane. An interesting aspect of the problem is that such traveling waves would formally correspond to solutions bifurcating from an eigenvalue of infinite multiplicity, and we have used ideas from bifurcation theory to develop an expansion procedure for waves of small amplitude. We study numerically hyperelastic materials where the traveling wave problem also has a variational structure, and we see evidence of solutions with elastic displacements that have discontinuous derivative at the boundary.