This paper presents a high-quality human motion prediction method that accurately predicts future human poses given observed ones. Our method is based on the observation that a good "initial guess" of the future poses is very helpful in improving the forecasting accuracy. This motivates us to propose a novel two-stage prediction framework, including an init-prediction network that just computes the good guess and then a formal-prediction network that predicts the target future poses based on the guess. More importantly, we extend this idea further and design a multi-stage prediction framework where each stage predicts initial guess for the next stage, which brings more performance gain. To fulfill the prediction task at each stage, we propose a network comprising Spatial Dense Graph Convolutional Networks (S-DGCN) and Temporal Dense Graph Convolutional Networks (T-DGCN). Alternatively executing the two networks helps extract spatiotemporal features over the global receptive field of the whole pose sequence. All the above design choices cooperating together make our method outperform previous approaches by large margins: 6%-7% on Human3.6M, 5%-10% on CMU-MoCap, and 13%-16% on 3DPW. Code is available at https://github.com/705062791/PGBIG.