This paper proposes AmbientGAN, which trains a generative adversarial network using partial or noisy observations rather than fully observed samples. AmbientGAN trains the discriminator on the measurement domain rather than the raw data domain, allowing the generator to be trained without needing large amounts of good training data. The paper proves it is theoretically possible to recover the original data distribution even when the measurement process is not invertible. It presents experimental results showing AmbientGAN can generate high quality samples and recover the underlying data distribution from various types of lossy and noisy measurements.