It is well-known that deep neural networks (DNNs) are susceptible to adversarial attacks, exposing a severe fragility of deep learning systems. As the result, adversarial training (AT) method, by incorporating adversarial examples during training, represents a natural and effective approach to strengthen the robustness of a DNN-based classifier. However, most AT-based methods, notably PGD-AT and TRADES, typically seek a pointwise adversary that generates the worst-case adversarial example by independently perturbing each data sample, as a way to ``probe'' the vulnerability of the classifier. Arguably, there are unexplored benefits in considering such adversarial effects from an entire distribution. To this end, this paper presents a unified framework that connects Wasserstein distributional robustness with current state-of-the-art AT methods. We introduce a new Wasserstein cost function and a new series of risk functions, with which we show that standard AT methods are special cases of their counterparts in our framework. This connection leads to an intuitive relaxation and generalization of existing AT methods and facilitates the development of a new family of distributional robustness AT-based algorithms. Extensive experiments show that our distributional robustness AT algorithms robustify further their standard AT counterparts in various settings.