The confluence of the advancement of Autonomous Vehicles (AVs) and the maturity of Vehicle-to-Everything (V2X) communication has enabled the capability of cooperative connected and automated vehicles (CAVs). Building on top of cooperative perception, this paper explores the feasibility and effectiveness of cooperative motion prediction. Our method, CMP, takes LiDAR signals as input to enhance tracking and prediction capabilities. Unlike previous work that focuses separately on either cooperative perception or motion prediction, our framework, to the best of our knowledge, is the first to address the unified problem where CAVs share information in both perception and prediction modules. Incorporated into our design is the unique capability to tolerate realistic V2X bandwidth limitations and transmission delays, while dealing with bulky perception representations. We also propose a prediction aggregation module, which unifies the predictions obtained by different CAVs and generates the final prediction. Through extensive experiments and ablation studies, we demonstrate the effectiveness of our method in cooperative perception, tracking, and motion prediction tasks. In particular, CMP reduces the average prediction error by 17.2\% with fewer missing detections compared with the no cooperation setting. Our work marks a significant step forward in the cooperative capabilities of CAVs, showcasing enhanced performance in complex scenarios.