The widely deployed ways to capture a set of unorganized points, e.g., merged laser scans, fusion of depth images, and structure-from- x , usually yield a 3-D noisy point cloud. Accurate normal estimation for the noisy point cloud makes a crucial contribution to the success of various applications. However, the existing normal estimation wisdoms strive to meet a conflicting goal of simultaneously performing normal filtering and preserving surface features, which inevitably leads to inaccurate estimation results. We propose a normal estimation neural network (Norest-Net), which regards normal filtering and feature preservation as two separate tasks, so that each one is specialized rather than traded off. For full noise removal, we present a normal filtering network (NF-Net) branch by learning from the noisy height map descriptor (HMD) of each point to the ground-truth (GT) point normal; for surface feature recovery, we construct a normal refinement network (NR-Net) branch by learning from the bilaterally defiltered point normal descriptor (B-DPND) to the GT point normal. Moreover, NR-Net is detachable to be incorporated into the existing normal estimation methods to boost their performances. Norest-Net shows clear improvements over the state of the arts in both feature preservation and noise robustness on synthetic and real-world captured point clouds.