In this paper, we propose a Point Fractal Network (PF-Net), a novel learning-based approach for precise and high-fidelity point cloud completion. Unlike existing point cloud completion networks, which generate the overall shape of the point cloud from the incomplete point cloud and always change existing points and encounter noise and geometrical loss, PF-Net preserves the spatial arrangements of the incomplete point cloud and can figure out the detailed geometrical structure of the missing region(s) in the prediction. To succeed at this task, PF-Net estimates the missing point cloud hierarchically by utilizing a feature-points-based multi-scale generating network. Further, we add up multi-stage completion loss and adversarial loss to generate more realistic missing region(s). The adversarial loss can better tackle multiple modes in the prediction. Our experiments demonstrate the effectiveness of our method for several challenging point cloud completion tasks.