Beyond local convolution networks, we explore how to harness various external human knowledge for endowing the networks with the capability of semantic global reasoning. Rather than using separate graphical models (e.g. CRF) or constraints for modeling broader dependencies, we propose a new Symbolic Graph Reasoning (SGR) layer, which performs reasoning over a group of symbolic nodes whose outputs explicitly represent different properties of each semantic in a prior knowledge graph. To cooperate with local convolutions, each SGR is constituted by three modules: a) a primal local-to-semantic voting module where the features of all symbolic nodes are generated by voting from local representations; b) a graph reasoning module propagates information over knowledge graph to achieve global semantic coherency; c) a dual semantic-to-local mapping module learns new associations of the evolved symbolic nodes with local representations, and accordingly enhances local features. The SGR layer can be injected between any convolution layers and instantiated with distinct prior graphs. Extensive experiments show incorporating SGR significantly improves plain ConvNets on three semantic segmentation tasks and one image classification task. More analyses show the SGR layer learns shared symbolic representations for domains/datasets with the different label set given a universal knowledge graph, demonstrating its superior generalization capability.