Generative adversarial networks (GANs) are often billed as "universal\ndistribution learners", but precisely what distributions they can represent and\nlearn is still an open question. Heavy-tailed distributions are prevalent in\nmany different domains such as financial risk-assessment, physics, and\nepidemiology. We observe that existing GAN architectures do a poor job of\nmatching the asymptotic behavior of heavy-tailed distributions, a problem that\nwe show stems from their construction. Additionally, when faced with the\ninfinite moments and large distances between outlier points that are\ncharacteristic of heavy-tailed distributions, common loss functions produce\nunstable or near-zero gradients. We address these problems with the Pareto GAN.\nA Pareto GAN leverages extreme value theory and the functional properties of\nneural networks to learn a distribution that matches the asymptotic behavior of\nthe marginal distributions of the features. We identify issues with standard\nloss functions and propose the use of alternative metric spaces that enable\nstable and efficient learning. Finally, we evaluate our proposed approach on a\nvariety of heavy-tailed datasets.\n