Abstract: We study the problem of robust distribution estimation under the Wasserstein metric, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. We introduce a new outlier-robust Wasserstein distance Wεp which allows for ε outlier mass to be removed from its input distributions, and show that minimum distance estimation under Wεp achieves minimax optimal robust estimation risk. Our analysis is rooted in several new results for partial OT, including an approximate triangle inequality, which may be of independent interest. To address computational tractability, we derive a dual formulation for Wεp that adds a simple penalty term to the classic Kantorovich dual objective. As such, Wεp can be implemented via an elementary modification to standard, duality-based OT solvers.