As real-scanned point clouds are mostly partial due to occlusions and viewpoints, reconstructing complete 3D shapes based on incomplete observations becomes a fundamental problem for computer vision. With a single incomplete point cloud, it becomes the partial point cloud completion problem. Given multiple different observations, 3D reconstruction can be addressed by performing partial-to-partial point cloud registration. Recently, a large-scale Multi-View Partial (MVP) point cloud dataset has been released, which consists of over 100,000 high-quality virtual-scanned partial point clouds. Based on the MVP dataset, this paper reports methods and results in the Multi-View Partial Point Cloud Challenge 2021 on Completion and Registration. In total, 128 participants registered for the competition, and 31 teams made valid submissions. The top-ranked solutions will be analyzed, and then we will discuss future research directions.