Abstract: To assess the difference between real and synthetic data, Generative Adversarial Networks (GANs) are trained using a distribution discrepancy measure. Three widely employed measures are information-theoretic divergences, integral probability metrics, and Hilbert space discrepancy metrics. We elucidate the theoretical connections between these three popular GAN training criteria and propose a novel procedure, called $\chi^2$ (Chi-square) GAN, that is conceptually simple, stable at training and resistant to mode collapse. Our procedure naturally generalizes to address the problem of simultaneous matching of multiple distributions. Further, we propose a resampling strategy that significantly improves sample quality, by repurposing the trained critic function via an importance weighting mechanism. Experiments show that the proposed procedure improves stability and convergence, and yields state-of-art results on a wide range of generative modeling tasks.