The problem of a nonlinearly elastic half-space and slab subjected to a steadily moving line force on the boundary is investigated. The conditions for the hyperbolicity of the governing equations are derived and shown to be dependent on the deformation gradients, particle velocities and the force speed. A numerical method of solution is applied whose stability criteria are derived and is shown to yield satisfactory accurate solutions. The conditions for the existence of generalized simple waves in the half-space are obtained and applied to produce primary dilatational and primary transverse simple waves. It is shown that an apparent anisotropy is introduced by the travelling force in the sense that the nonlinear response and velocities generated by a positive tangential force is different from that of a negative one. For a moving force over the surface of a slab, it is shown that waves propagate in opposite directions, from a point within the slab, at different velocities.