This text constitutes a revised edition of the book published in French by the ̀Presses Polytechniques et Universitaires Romandes' in 1991. It contains a comprehensive treatment of the theory of random vibration of linear structures subjected to stationary or nonstationary excitations. The book consists of three parts: Chapters 1 to 4 review the theory of random processes; Chapters 5 and 6 analyse the response of linear structures to various classes of random excitations. Illustrative examples are taken from various fields of structural engineering, such as boundary layer noise, along wind response of buildings or the seismic response of multi-supported structures. The spectral description of nonstationary oscillatory processes and the response of linear structures to nonstationary excitation are treated in Chapter 8. The third part is devoted to the failure modes by threshold crossing and random fatigue; classical results for fatigue under uniaxial loading are extended to biaxial stress fields. The final chapter is devoted to the discrete Fourier transform. This book will interest structural analysts and research engineers dealing with structures in a random environment. It can also be used as a textbook for a graduate course in random vibration. The text is supplemented by 81 problems.