Title: Outlier Robust Optimal Transport. Abstract: Optimal transport (OT) provides a way of measuring distances between distributions that depends on the geometry of the sample space. In light of recent advances in solving the OT problem, OT distances are widely used as loss functions in minimum distance estimation. Despite its prevalence and advantages, however, OT is extremely sensitive to outliers. A single adversarially-picked outlier can increase OT distance arbitrarily. To address this issue, in this work we propose an outlier-robust OT formulation. Our formulation is convex but challenging to scale at a first glance. We proceed by deriving an equivalent formulation based on cost truncation that is easy to incorporate into modern stochastic algorithms for regularized OT. We demonstrate our model applied to mean estimation under the Huber contamination model in simulation as well as outlier detection on real data.