Motivated by the successes of deep learning, we propose a class of neural network-based discrete choice models, called RUMnets, which is inspired by the random utility maximization (RUM) framework. This model formulates the agents’ random utility function using the sample average approximation (SAA) method. We show that RUMnets sharply approximate the class of RUM discrete choice models: any model derived from random utility maximization has choice probabilities that can be approximated arbitrarily closely by a RUMnet. Reciprocally, any RUMnet is consistent with the RUM principle. We derive an upper bound on the generalization error of RUMnets fitted on choice data, and gain theoretical insights on their ability to predict choices on new, unseen data depending on critical parameters of the dataset and architecture.  By leveraging open-source libraries for neural networks, we find that RUMnets outperform other state-of-the-art choice modeling and machine learning methods by a significant margin on two real-world datasets. This is joint work with Ali Aouad (LBS).