Nonlinear stochastic dynamics is a broad topic well beyond the scope of this book. Chapter 13 describes a particular method of solution for a certain class of nonlinear stochastic dynamic problem by use of FORM. The approach belongs to the class of solution methods known as equivalent linearization. In this case, the linearization is carried out by replacing the nonlinear system with a linear one that has a tail probability equal to the FORM approximation of the tail probability of the nonlinear system – hence the name tail-equivalent linearization method. The equivalent linear system is obtained non-parametrically in terms of its unit impulse response function. For small failure probabilities, the accuracy of the method is shown to be far superior to those of other linearization methods. Furthermore, the method is able to capture the non-Gaussian distribution of the nonlinear response. This chapter develops this method for systems subjected to Gaussian and non-Gaussian excitations and nonlinear systems with differentiable loading paths. Approximations for level crossing rates and the first-passage probability are also developed. The method is extended to nonlinear structures subjected to multiple excitations, such as bi-component base motion, and to evolutionary input processes.