Wasserstein distributionally robust optimization (DRO) estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by the need to identify model parameters (or) decision choices that are robust to model uncertainties and misspecification, the Wasserstein DRO estimators recover a wide range of regularized estimators, including square-root LASSO and support vector machines, among others, as particular cases. This paper studies the asymptotic normality of underlying DRO estimators as well as the properties of an optimal (in a suitable sense) confidence region induced by the Wasserstein DRO formulation.