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We introduce the first neural optimization framework to solve a classical instance of the tiling problem. Namely, we seek a non-periodic tiling of an arbitrary 2D shape using one or more types of tiles: the tiles maximally fill the shape's interior without overlaps or holes. To start, we reformulate tiling as a graph problem by modeling candidate tile locations in the target shape as graph nodes and connectivity between tile locations as edges. Further, we build a graph convolutional neural network, coined TilinGNN, to progressively propagate and aggregate features over graph edges and predict tile placements. TilinGNN is trained by maximizing the tiling coverage on target shapes, while avoiding overlaps and holes between the tiles. Importantly, our network is self-supervised, as we articulate these criteria as loss terms defined on the network outputs, without the need of ground-truth tiling solutions. After training, the runtime of TilinGNN is roughly linear to the number of candidate tile locations, significantly outperforming traditional combinatorial search. We conducted various experiments on a variety of shapes to showcase the speed and versatility of TilinGNN. We also present comparisons to alternative methods and manual solutions, robustness analysis, and ablation studies to demonstrate the quality of our approach.