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Point cloud filtering is a fundamental problem in geometry modeling and processing. Despite of significant advancement in recent years, the existing methods still suffer from two issues: 1) they are either designed without preserving sharp features or less robust in feature preservation; and 2) they usually have many parameters and require tedious parameter tuning. In this paper, we propose a novel deep learning approach that automatically and robustly filters point clouds by removing noise and preserving their sharp features. Our point-wise learning architecture consists of an encoder and a decoder. The encoder directly takes points (a point and its neighbors) as input, and learns a latent representation vector which goes through the decoder to relate the ground-truth position with a displacement vector. The trained neural network can automatically generate a set of clean points from a noisy input. Extensive experiments show that our approach outperforms the state-of-the-art deep learning techniques in terms of both visual quality and quantitative error metrics. The source code and dataset can be found at https://github.com/dongbo-BUAA-VR/Pointfilter.