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 features. In particular, we leverage discrete graph curvature, which measures how the neighborhoods of a pair of nodes are structurally related. The curvature of an edge (x, y) defines the distance taken to travel from neighbors of x to neighbors of y, compared with the length of edge (x, y). It is a much more descriptive feature compared to previously used features that only focus on node specific attributes or limited topological information such as degree. Our curvature graph convolution network outperforms state-of-the-art on various synthetic and real-world graphs, especially the larger and denser ones.