In the last few years, generative adversarial networks (GAN) have shown tremendous potential for a number of applications in computer vision and related fields. With the current pace of progress, it is a sure bet they will soon be able to generate high-quality images and videos, virtually indistinguishable from real ones. Unfortunately, realistic GAN-generated images pose serious threats to security, to begin with a possible flood of fake multimedia, and multimedia forensic countermeasures are in urgent need. In this work, we show that each GAN leaves its specific fingerprint in the images it generates, just like real-world cameras mark acquired images with traces of their photo-response non-uniformity pattern. Source identification experiments with several popular GANs show such fingerprints to represent a precious asset for forensic analyses.