Title: Propagating Distributions through Neural Networks. Abstract: We propose a new approach to propagating probability distributions through neural networks. To handle non-linearities, we use local linearization and show this to be an optimal approximation in terms of total variation for ReLUs. We demonstrate the advantages of our method over the moment matching approach popularized in prior works. In addition, we formulate new loss functions for training neural networks based on distributions. To demonstrate the utility of propagating distributions, we apply it to quantifying prediction uncertainties. In regression tasks we obtain calibrated confidence intervals, and in a classification setting we improve selective prediction on out-of-distribution data. We also show empirically that training with our uncertainty aware losses improve robustness to random and adversarial noise.