We consider structure discovery of undirected graphical models from observational data. Inferring likely structures from few examples is a complex task often requiring the formulation of priors and sophisticated inference procedures. In the setting of Gaussian Graphical Models (GGMs) a popular estimator is a penalized maximum likelihood objective on the precision matrix. Adapting this objective to capture domain-specific knowledge as priors or a new data likelihood requires great effort. In addition, structure recovery is a very indirect consequence of the data-fit term. By contrast, it may be easier to generate training samples of data that arise from graphs with the desired properties. We propose here to leverage this latter source of information in order to learn a function that maps from empirical covariance matrices to estimated graph structures. Learning this function brings two benefits: it implicitly models the desired structure or sparsity properties to form suitable priors, and it can more directly be tailored to the specific problem of edge structure discovery. We apply this framework to several critical real world problems in structure discovery and show that it can be competitive to standard approaches such as graphical lasso, at a fraction of the execution speed. We use convolutional neural networks to parametrize our estimators due to the compositional block structure of matrix inversion. Experimentally, our learnable graph-discovery method trained on synthetic data generalizes well to different data: identifying relevant edges in real data, completely unknown at training time. We find that on genetics, brain imaging, and simulation data we obtain competitive (and often superior) performance, compared with analytical methods.