In this paper, we propose a robust and efficient end-to-end non-local spatial propagation network for depth completion. The proposed network takes RGB and sparse depth images as inputs and estimates non-local neighbors and their affinities of each pixel, as well as an initial depth map with pixel-wise confidences. The initial depth prediction is then iteratively refined by its confidence and non-local spatial propagation procedure based on the predicted non-local neighbors and corresponding affinities. Unlike previous algorithms that utilize fixed-local neighbors, the proposed algorithm effectively avoids irrelevant local neighbors and concentrates on relevant non-local neighbors during propagation. In addition, we introduce a learnable affinity normalization to better learn the affinity combinations compared to conventional methods. The proposed algorithm is inherently robust to the mixed-depth problem on depth boundaries, which is one of the major issues for existing depth estimation/completion algorithms. Experimental results on indoor and outdoor datasets demonstrate that the proposed algorithm is superior to conventional algorithms in terms of depth completion accuracy and robustness to the mixed-depth problem. Our implementation is publicly available on the project page ( https://github.com/zzangjinsun/NLSPN_ECCV20 ).