Structural reliability analysis aims at computing the probability of failure of systems whose performance may be assessed by using complex computational models (e.g., expensive-to-run finite-element models). A direct use of Monte Carlo simulation is not feasible in practice, unless a surrogate model (such as kriging, also known as Gaussian process modeling) is used. Such metamodels are often used in conjunction with adaptive experimental designs (i.e., design enrichment strategies), which allows one to iteratively increase the accuracy of the surrogate for the estimation of the failure probability while keeping low the overall number of runs of the costly original model. This paper develops a new structural reliability method based on the recently developed polynomial-chaos kriging (PC-kriging) approach coupled with an active learning algorithm known as adaptive kriging Monte Carlo simulation (AK-MCS). The problem is formulated in such a way that the computation of both small probabilities of failure and extreme quantiles is unified. Different convergence criteria for both types of analyses are discussed, and in particular the original AK-MCS stopping criterion is shown to be overconservative. A multipoint enrichment algorithm is elaborated, which allows the addition of several points in each iteration, thus fully exploiting high-performance computing architectures. The proposed method is illustrated on three examples, namely a two-dimensional case that allows underlining of the advantages of this approach compared to standard AK-MCS. Then the quantiles of the eight-dimensional borehole function are estimated. Finally the reliability of a truss structure (10 random variables) is addressed. In all cases, accurate results are obtained with approximately 100 runs of the original model.