Graph Quilting Graphical model estimation is a seemingly impossible task when several pairs of variables are not observed jointly. Recovering the edges of the graph in such settings requires one to infer conditional dependencies between variables with no evidence of their marginal dependence. This unexplored statistical problem arises in several situations, such as in large-scale neuroimaging where, because of technology limitations, it is impossible to record the activities of thousands of neurons simultaneously. We call this statistical challenge the "Graph Quilting problem". We study this problem for Gaussian Graphical models and first show that, under mild conditions, it is possible to correctly identify edges connecting the observed pairs of nodes. Additionally, we show that we can recover a minimal superset of the edges connecting variables that are never jointly observed. Thus, we show that one can infer conditional relationships even when marginal relationships are unknown. To accomplish this, we devise a novel technique that we call the "Recursive-Complement" algorithm. We propose an L1-regularized graph quilting estimator and establish its rates of convergence for graph estimation and selection in high-dimensions. We illustrate our approach using synthetic data, as well as data obtained from in vivo calcium imaging of ten thousand neurons in mouse visual cortex. We further discuss several other applications.