Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first-order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P-wave velocity to S-wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite-difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.