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Graph convolutional network based methods that model the body-joints' relations, have recently shown great promise in 3D skeleton-based human motion prediction. However, these methods have two critical issues: first, deep graph convolutions filter features within only limited graph spectrums, losing sufficient information in the full band; second, using a single graph to model the whole body underestimates the diverse patterns on various body-parts. To address the first issue, we propose adaptive graph scattering, which leverages multiple trainable band-pass graph filters to decompose pose features into richer graph spectrum bands. To address the second issue, body-parts are modeled separately to learn diverse dynamics, which enables finer feature extraction along the spatial dimensions. Integrating the above two designs, we propose a novel skeleton-parted graph scattering network (SPGSN). The cores of the model are cascaded multi-part graph scattering blocks (MPGSBs), building adaptive graph scattering on diverse body-parts, as well as fusing the decomposed features based on the inferred spectrum importance and body-part interactions. Extensive experiments have shown that SPGSN outperforms state-of-the-art methods by remarkable margins of 13.8%, 9.3% and 2.7% in terms of 3D mean per joint position error (MPJPE) on Human3.6M, CMU Mocap and 3DPW datasets, respectively.