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Raw point clouds data inevitably contains outliers or noise through acquisition from 3D sensors or reconstruction algorithms. In this paper, we present a novel end-to-end network for robust point clouds processing, named PointASNL, which can deal with point clouds with noise effectively. The key component in our approach is the adaptive sampling (AS) module. It first re-weights the neighbors around the initial sampled points from farthest point sampling (FPS), and then adaptively adjusts the sampled points beyond the entire point cloud. Our AS module can not only benefit the feature learning of point clouds, but also ease the biased effect of outliers. To further capture the neighbor and long-range dependencies of the sampled point, we proposed a local-nonlocal (L-NL) module inspired by the nonlocal operation. Such L-NL module enables the learning process insensitive to noise. Extensive experiments verify the robustness and superiority of our approach in point clouds processing tasks regardless of synthesis data, indoor data, and outdoor data with or without noise. Specifically, PointASNL achieves state-of-the-art robust performance for classification and segmentation tasks on all datasets, and significantly outperforms previous methods on real-world outdoor SemanticKITTI dataset with considerate noise. Our code is released through https://github.com/yanx27/PointASNL.