3D skeleton-based action recognition and motion prediction are two essential problems of human activity understanding. In many previous works: 1) they studied two tasks separately, neglecting internal correlations; 2) they did not capture sufficient relations inside the body. To address these issues, we propose a symbiotic model to handle two tasks jointly; and we propose two scales of graphs to explicitly capture relations among body-joints and body-parts. Together, we propose symbiotic graph neural networks, which contain a backbone, an action-recognition head, and a motion-prediction head. Two heads are trained jointly and enhance each other. For the backbone, we propose multi-branch multi-scale graph convolution networks to extract spatial and temporal features. The multi-scale graph convolution networks are based on joint-scale and part-scale graphs. The joint-scale graphs contain actional graphs, capturing action-based relations, and structural graphs, capturing physical constraints. The part-scale graphs integrate body-joints to form specific parts, representing high-level relations. Moreover, dual bone-based graphs and networks are proposed to learn complementary features. We conduct extensive experiments for skeleton-based action recognition and motion prediction with four datasets, NTU-RGB+D, Kinetics, Human3.6M, and CMU Mocap. Experiments show that our symbiotic graph neural networks achieve better performances on both tasks compared to the state-of-the-art methods.