A mathematical statement of the problem of optimum design of shock isolation systems is presented and two computational methods of solution are examined. These are termed the direct and indirect methods of design optimization. The direct method is the more conventional, and employs numerical search procedures common to problems of constrained minimization. As such, its main limitations stem from the usual computational burden associated with large problems. The indirect method is akin to modern techniques of control theory, and appears applicable to large, complex systems; moreover, it provides information useful to the designer on the bounds of possible system performance that is not available through the direct method. An extensive bibliography is presented.