Deep neural networks on 3D point cloud data have been widely used in the real world, especially in safety-critical applications. However, their robustness against corruptions is less studied. In this paper, we present ModelNet40-C, the first comprehensive benchmark on 3D point cloud \textit{corruption robustness}, consisting of 15 common and realistic corruptions. Our evaluation shows a significant gap between the performances on ModelNet40 and ModelNet40-C for state-of-the-art (SOTA) models. We also demonstrate the effectiveness of different data augmentation strategies in enhancing robustness for different corruption types. We hope our in-depth analysis will motivate the development of robust training strategies or architecture designs in the 3D point cloud domain. Our codebase and dataset will be made available upon acceptance.