Real-time 3D object detection is a fundamental technique in numerous applications, such as autonomous driving, unmanned aerial vehicles (UAV) and robot vision. However, current LiDAR-based 3D object detection algorithms allocate inadequate attention to the inhomogeneity of LiDAR point clouds and the shape encoding capability of regional proposal schemes. This paper introduces a novel 3D object detection network called the Shape Attention Regional Proposal Network (SARPNET), which deploys a new low-level feature encoder to remedy the sparsity and inhomogeneity of LiDAR point clouds with an even sample method, and embodies a shape attention mechanism that learns the statistic 3D shape priors of objects and uses them to spatially enhance semantic embeddings. Experimental results show that the proposed one-stage method outperforms state-of-the-art one-stage and even two-stage methods on the KITTI 3D object detection benchmark. It achieved a BEV AP of (87.26%, 62.80%), 3D AP of (75.64%, 60.43%), and orientation AP of (88.86%, 71.01%) for the detection of cars and cyclists, respectively. Besides, the method is the third winner in the nuScenes 3D Detection challenge of CVPR2019 Workshop on Autonomous Driving (WAD).