Networks have become instrumental in deciphering how information is processed and transferred within systems in almost every scientific field today. Nearly all network analyses, however, have relied on humans to devise structural features of networks believed to be most discriminative for an application. We present a framework for comparing and classifying networks without human-crafted features using deep learning. After training, autoencoders contain hidden units that encode a robust structural vocabulary for succinctly describing graphs. We use this feature vocabulary to tackle several network mining problems and find improved predictive performance versus many popular features used today. These problems include uncovering growth mechanisms driving the evolution of networks, predicting protein network fragility, and identifying environmental niches for metabolic networks. Deep learning offers a principled approach for mining complex networks and tackling graph-theoretic problems.