Abstract:  In many areas of science and engineering, it is of interest to estimate rare event probabilities. This is particularly relevant in reliability analysis and risk management of engineering systems, wherein estimation of failure probabilities is essential. In this context, the rare event of interest is often defined point-wise in terms of the outcome of a computationally intensive numerical model. Rare event probabilities are often estimated with Monte Carlo-based sampling approaches due to their robustness with respect to the complexity of the underlying model. Although the performance of the Monte Carlo method does not depend on the dimension of the random variable space, it becomes poor when the target probability is small. In this talk, a number of advanced sampling methods are discussed that improve the efficiency of crude Monte Carlo, while maintaining to a certain extent its independency on the number of random variables. In particular, we discuss subset simulation, sequential importance sampling and the cross entropy method. These methods perform a series of sampling steps with aim at obtaining samples from a theoretically optimal importance sampling density. The performance of the methods is demonstrated with numerical examples in both low and high dimensional component and system reliability problems.