Structural reliability theory has been applied to many engineering problems in the last decades, with the primary objective of quantifying the safety of such structures. Although in some cases approximated methods may be used, many times the only alternatives are those involving more demanding approaches, such as Monte Carlo simulation (MCS). In this context, surrogate models have been widely employed as an attempt to keep the computational effort acceptable. In this paper, an adaptive approach for reliability analysis using surrogate models, proposed in the literature in the context of Kriging and polynomial chaos expansions (PCEs), is adapted for the case of multilayer perceptron (MLP) artificial neural networks (ANNs). The methodology is employed in the solution of three benchmark reliability problems and compared to MCS and other methods from the literature. In all cases, the ANNs led to results very close to those obtained by MCS and required much less limit state function evaluations. Also, the performance of the ANNs was found comparable, in terms of accuracy and efficiency, to the performance of the other methods.