Graph-structured data is prevalent in many domains. Despite the widely celebrated success of deep neural networks, their power in graph-structured data is yet to be fully explored. We propose a novel network architecture that incorporates advanced graph structural information, specifically, discrete graph curvature, which measures how the neighborhoods of a pair of nodes are structurally related. The curvature of an edge (x, y) is defined by comparing the distance taken to travel from neighbors of x to neighbors of y, with the length of edge (x, y). It is a much more descriptive structural measure compared to previously ones that only focus on node specific attributes or limited graph topological information such as degree. Our curvature graph convolution network outperforms state-of-the-art methods on various synthetic and real-world graphs, especially the large and dense ones.