In a probabilistic assessment of the performance of structures subjected to uncertain environmental loads such as earthquakes, an important problem is to determine the probability that the structural response exceeds some specified limits within a given duration of interest. This problem is known as the first excursion problem, and it has been a challenging problem in the theory of stochastic dynamics and reliability analysis. In spite of the enormous amount of attention the problem has received, there is no procedure available for its general solution, especially for engineering problems of interest where the complexity of the system is large and the failure probability is small. The application of simulation methods to solving the first excursion problem is investigated in this dissertation, with the objective of assessing the probabilistic performance of structures subjected to uncertain earthquake excitations modeled by stochastic processes. From a simulation perspective, the major difficulty in the first excursion problem comes from the large number of uncertain parameters often encountered in the stochastic description of the excitation. Existing simulation tools are examined, with special regard to their applicability in problems with a large number of uncertain parameters. Two efficient simulation methods are developed to solve the first excursion problem. The first method is developed specifically for linear dynamical systems, and it is found to be extremely efficient compared to existing techniques. The second method is more robust to the type of problem, and it is applicable to general dynamical systems. It is efficient for estimating small failure probabilities because the computational effort grows at a much slower rate with decreasing failure probability than standard Monte Carlo simulation. The simulation methods are applied to assess the probabilistic performance of structures subjected to uncertain earthquake excitation. Failure analysis is also carried out using the samples generated during simulation, which provide insight into the probable scenarios that will occur given that a structure fails.