Deep neural networks (DNNs) are known to be vulnerable to adversarial attacks that would trigger misclassification of DNNs but may be imperceptible to human perception. Adversarial defense has been an important way to improve the robustness of DNNs. Existing attack methods often construct adversarial examples relying on some metrics like the $\ell_p$ distance to perturb samples. However, these metrics can be insufficient to conduct adversarial attacks due to their limited perturbations. In this paper, we propose a new internal Wasserstein distance (IWD) to capture the semantic similarity of two samples, and thus it helps to obtain larger perturbations than currently used metrics such as the $\ell_p$ distance. We then apply the internal Wasserstein distance to perform adversarial attack and defense. In particular, we develop a novel attack method relying on IWD to calculate the similarities between an image and its adversarial examples. In this way, we can generate diverse and semantically similar adversarial examples that are more difficult to defend by existing defense methods. Moreover, we devise a new defense method relying on IWD to learn robust models against unseen adversarial examples. We provide both thorough theoretical and empirical evidence to support our methods.