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This paper reviews the NTIRE 2020 challenge on video quality mapping (VQM), which addresses the issues of quality mapping from source video domain to target video domain. The challenge includes both a supervised track (track 1) and a weakly-supervised track (track 2) for two benchmark datasets. In particular, track 1 offers a new Internet video benchmark, requiring algorithms to learn the map from more compressed videos to less compressed videos in a supervised training manner. In track 2, algorithms are required to learn the quality mapping from one device to another when their quality varies substantially and weakly-aligned video pairs are available. For track 1, in total 7 teams competed in the final test phase, demonstrating novel and effective solutions to the problem. For track 2, some existing methods are evaluated, showing promising solutions to the weakly-supervised video quality mapping problem.