Neural networks are becoming increasingly prevalent in software, and it is therefore important to be able to verify their behavior. Because verifying the correctness of neural networks is extremely challenging, it is common to focus on the verification of other properties of these systems. One important property, in particular, is robustness. Most existing definitions of robustness, however, are at one of two extremes:practical definitions, whose notion of robustness is too weak to be useful, or (too) strong definitions that are unlikely to be satisfied by—and verifiable for—practical neural networks. To strike a balance between these two extremes, we propose a novel notion of robustness: probabilistic robustness. Given a probability distribution over the inputs to a neural network, probabilistic robustness requires the neural network to be robust with at least (1 − \epsilon) probability, This probabilistic approach is practical and provides a principled way of estimating the robustness of a neural network. We also present an algorithm, based on abstract interpretation and importance sampling, for checking whether a neural network is probabilistically robust. Our algorithm uses abstract interpretation to approximate the behavior of a neural network and compute an overapproximation of the input regions that violate robustness. It then uses importance sampling to counter the effect of such overapproximation and compute an accurate estimate of the probability that the neural network violates the robustness property.