Wasserstein GANs are based on the idea of minimising the Wasserstein distance\nbetween a real and a generated distribution. We provide an in-depth\nmathematical analysis of differences between the theoretical setup and the\nreality of training Wasserstein GANs. In this work, we gather both theoretical\nand empirical evidence that the WGAN loss is not a meaningful approximation of\nthe Wasserstein distance. Moreover, we argue that the Wasserstein distance is\nnot even a desirable loss function for deep generative models, and conclude\nthat the success of Wasserstein GANs can in truth be attributed to a failure to\napproximate the Wasserstein distance.\n