The principal criteria for the design of hammer foundations are specified and the methods of defining stiffness and damping constants for shallow foundations and pile foundations are reviewed. An analytical approach to the prediction of damped vibration is presented. The incorporation of damping is based on an energy consideration and verified by comparison with results obtained using the complex eigenvalue approach. The inclusion of damping makes the response prediction more realistic and may prevent considerable overestimation of the response and overdesign of the foundation. Keywords: damping, dynamics, foundations, hammers, shock, vibration.