Abstract: The first passage problems of linear and nonlinear dynamical systems excited by Gauss white noise are considered. For linear dynamical system, the failure domain can be described as a union of mutually exclusive events, and every event is completely described by a local design point. The paper uses standard Gaussian distribution instead of chi-square distribution to estimate the parameter of first passage probability. For nonlinear dynamical system, the equivalent linear system is carried out based on the out-crossing theory. The linearization principle is that nonlinear and linear systems have the same up-crossing rate for a specified threshold. Finally the paper gives two examples. The results show that the method of the paper suggested is correct and effective by comparing with the Monte Carlo method.