With the representation effectiveness, skeleton-based human action recognition has received considerable research attention, and has a wide range of real applications. In this area, many existing methods typically rely on fixed physicalconnectivity skeleton structure for recognition, which is incapable of well capturing the intrinsic high-order correlations among skeleton joints. In this paper, we propose a novel spatio-temporal graph routing (STGR) scheme for skeletonbased action recognition, which adaptively learns the intrinsic high-order connectivity relationships for physicallyapart skeleton joints. Specifically, the scheme is composed of two components: spatial graph router (SGR) and temporal graph router (TGR). The SGR aims to discover the connectivity relationships among the joints based on sub-group clustering along the spatial dimension, while the TGR explores the structural information by measuring the correlation degrees between temporal joint node trajectories. The proposed scheme is naturally and seamlessly incorporated into the framework of graph convolutional networks (GCNs) to produce a set of skeleton-joint-connectivity graphs, which are further fed into the classification networks. Moreover, an insightful analysis on receptive field of graph node is provided to explain the necessity of our method. Experimental results on two benchmark datasets (NTU-RGB+D and Kinetics) demonstrate the effectiveness against the state-of-the-art.