This thesis addresses the application of random vibration theory to seismic engineering of piping in the nuclear industry. Using the time domain state space approach, random vibration modal time history (RVMTH) theory is developed for primary structures and cascaded secondary multiply supported piping (P-S) systems or other cascaded subsystems subject to high level earthquake loads. The ground excitations are modelled as non-stationary, non-white, correlated and vector-valued stochastic processes. The excitation components are defined by prescribed strength envelope functions, the RMS values of ground acceleration, and second order filter parameters that define frequency content and bandwidth. The effects of correlation between excitation components and between modal coordinates are explicitly included. The pseudo static effect concept is used to compute piping total responses. Time histories of RMS values, correlation coefficients and covariance matrices can be computed for any response vector of interest. The computer program SSRV-PC that implements the RVMTH formulation is linked with a commercial, general purpose finite element program to handle realistic models. The output of the RVMTH analysis is used to synthesize correlated, vector-valued, evolu tionary processes of elbow total displacements. Auto Regressive AR(p) models are employed. Response comparisons are made using the RVMTH and deterministic methods (modal time history, independent support motion, and multiple modal time history) for two models to investigate the uncertainties behind the deterministic methods. The first model is a 3D-frame with a fixed base and a rigidly designed piping system. The second model includes a full size four story shear wall building including structure-foundation interaction and a realistic flexible piping system that has four sizes of pipe. For application of the RVMTH approach, a design procedure is proposed that includes the state-of-the-art in random vibration theory and the NRC and ASME practices in the nuclear industry Advisors/Committee Members: Gasparini, Dario A.