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Image guided depth completion aims to recover per-pixel dense depth maps from sparse depth measurements with the help of aligned color images, which has a wide range of applications from robotics to autonomous driving. However, the 3D nature of sparse-to-dense depth completion has not been fully explored by previous methods. In this work, we propose a Graph Convolution based Spatial Propagation Network (GraphCSPN) as a general approach for depth completion. First, unlike previous methods, we leverage convolution neural networks as well as graph neural networks in a complementary way for geometric representation learning. In addition, the proposed networks explicitly incorporate learnable geometric constraints to regularize the propagation process performed in three-dimensional space rather than in two-dimensional plane. Furthermore, we construct the graph utilizing sequences of feature patches, and update it dynamically with an edge attention module during propagation, so as to better capture both the local neighboring features and global relationships over long distance. Extensive experiments on both indoor NYU-Depth-v2 and outdoor KITTI datasets demonstrate that our method achieves the state-of-the-art performance, especially when compared in the case of using only a few propagation steps. Code and models are available at the project page.