Robustness of deep learning models is a property that has recently gained increasing attention. We explore a notion of robustness for generative adversarial models that is pertinent to their internal interactive structure, and show that, perhaps surprisingly, the GAN in its original form is not robust. Our notion of robustness relies on a perturbed discriminator, or noisy, adversarial interference with its feedback. We explore, theoretically and empirically, the effect of model and training properties on this robustness. In particular, we show theoretical conditions for robustness that are supported by empirical evidence. We also test the effect of regularization. Our results suggest variations of GANs that are indeed more robust to noisy attacks and have more stable training behavior, requiring less regularization in general. Inspired by our theoretical results, we further extend our framework to obtain a class of models related to WGAN, with good empirical performance. Overall, our results suggest a new perspective on understanding and designing GAN models from the viewpoint of their internal robustness.