||This work is dedicated to the exploration of commonly used and development of new advanced stochastic simulation algorithms for solving high-dimensional reliability problems. Firstly, adopting a geometric point of view we highlight and explain a range of results concerning the performance of several reliability methods. Namely, we discuss Importance Sampling in high dimensions and provide a geometric understanding as to why Importance Sampling does generally 'not work' in high dimensions. We furthermore challenge the significance of 'design point' when dealing with strongly nonlinear problems. We conclude by showing that for the general high-dimensional nonlinear reliability problems the selection of an appropriate Importance Sampling density (ISD) is practically impossible. Also, we provide a geometric explanation as to why the standard Metropolis-Hastings (MH) algorithm does 'not work' in high-dimensions. Next, we develop two useful modifications of the well-known reliability methods. The first is Adaptive Linked Importance Sampling (ALIS), which generalizes Subset Simulation (SS) and in some cases can offer drastic improvements over SS. The second is Modified Metropolis-Hastings algorithm with Delayed Rejection (MMHDR) which is a novel modification of the MH algorithm, designed specially for sampling from conditional high-dimensional distributions. Finally, we propose a novel advanced stochastic simulation algorithm called Horseracing Simulation (HRS). The idea behind HS is the following. Although the reliability problem itself is high-dimensional, the limit-state function maps this high-dimensional parameter space into a one-dimensional real line. This mapping transforms a high-dimensional random parameter vector, which represents the input load, into a random variable with unknown distribution, which represents the structure response. It turns out, that the corresponding cumulative distribution function (CDF) of this random variable of interest can be accurately approximated by empirical CDFs constructed from specially designed samples. The accuracy and efficiency of the new method is demonstrated with a real-life wind engineering example.