This paper presents an approach to estimate probabilities of failure in the case of dependent random variables. The approach is based on copulas and support vector machines (SVMs). A copula is used to generate dependent Monte Carlo samples and an SVM is used to construct the explicit boundary of the failure domain. It is shown that this construction of the failure boundary cannot be made in the original space due to the lack of "isotropy" of the probability densities. In this work the SVM is built in the uncorrelated standard normal space and refined using an adaptive sampling scheme. A transformation is used to map SVM training points and Monte-Carlo samples between the original space and the uncorrelated standard normal space. Because SVM is a classification-based approach, it can handle discontinuous responses and, more importantly, several limit states using one single SVM. Several analytical examples are used to demonstrate the methodology.