Determination of the stresses and displacements which occur in response to random excitations cannot be accomplished by traditional deterministic analysis methods. As the specification of the excitation and the response of the structure become more complex, solutions by direct, closed-form methods require extensive computations. Two methods are presented which can be used in the analysis of structures which are subjected to random excitations. The Power Spectrum Method is a procedure which determines the random vibration response of the structure based upon a frequency response analysis of a structural model. The Response Spectrum Method is a method which is based upon specified forces or displacements as a function of time. A derivation of each of the methods is presented and followed by comparisons of the results which were obtained for single and multiple-degree-of-freedom systems. Assumptions and limitations of the methods are discussed as well as their accuracy over ranges of frequency, damping and loading specification. As a direct application and comparison of the two methods, an analysis of the support system for the primary mirror of the Space Infrared Telescope Facility (SIRTF) has been performed. In addition, a method for the evaluation of the critical damping in a single-degree-of-freedom structure is demonstrated.