Deep learning technique has yielded significant improvements in point cloud completion with the aim of completing missing object shapes from partial inputs. However, most existing methods fail to recover realistic structures due to over-smoothing of fine-grained details. In this paper, we develop a voxel-based network for point cloud completion by leveraging edge generation (VE-PCN). We first embed point clouds into regular voxel grids, and then generate complete objects with the help of the hallucinated shape edges. This decoupled architecture together with a multi-scale grid feature learning is able to generate more realistic on-surface details. We evaluate our model on the publicly available completion datasets and show that it outperforms existing state-of-the-art approaches quantitatively and qualitatively. Our source code is available at https://github.com/xiaogangw/VE-PCN.