The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained fromnindependent samples from μ approaches μ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radically different rates of convergence as n grows. See more details in: J. Weed, F. Bach. Sharp asymptotic and finite-sample ratesof convergence of empirical measures in Wasserstein distance. Technical Report, Arxiv-1707.00087, 2017.