We present a novel method for computing correspondences across 3D shapes using unsupervised learning. Our method computes a non-linear transformation of given descriptor functions, while optimizing for global structural properties of the resulting maps, such as their bijectivity or approximate isometry. To this end, we use the functional maps framework, and build upon the recent FMNet architecture for descriptor learning. Unlike that approach, however, we show that learning can be done in a purely \emph{unsupervised setting}, without having access to any ground truth correspondences. This results in a very general shape matching method that we call SURFMNet for Spectral Unsupervised FMNet, and which can be used to establish correspondences within 3D shape collections without any prior information. We demonstrate on a wide range of challenging benchmarks, that our approach leads to state-of-the-art results compared to the existing unsupervised methods and achieves results that are comparable even to the supervised learning techniques. Moreover, our framework is an order of magnitude faster, and does not rely on geodesic distance computation or expensive post-processing.