The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model, e.g. a GAN or a VAE. We show how the problems of image inpainting and super-resolution are special cases of our general framework. We show how to generalize the RIP condition for generative models and that random gaussian measurement matrices have this property with high probability. A Lipschitz condition for the generative neural network is a key technical condition. We will also discuss on-going work for adding using these GANs for defense against adversarial examples.  (Joint work with Ashish Bora, Ajil Jalal and Eric Price)