<p>We study a variety of Wasserstein distributionally robust optimization (WDRO) problems where the distributions in the ambiguity set are chosen by constraining their Wasserstein discrepancies to the empirical distribution. Using the notion of weak Lipschitz property, we develop a flexible framework to derive lower and upper bounds for the corresponding worst-case loss quantity and propose sufficient conditions under which this quantity coincides with its regularization scheme counterpart. Our constructive methodology and elementary analysis also directly characterize the closed-form of the approximate worst-case distribution. Extensive applications show that our framework can be used to establish theoretical results for various problems, including regression, classification and risk measure problems. We believe that the flexible framework developed here can serve as a convenient tool for analysing other WDRO problems.</p>