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For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors that occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of noisy intermediate-scale quantum machines one must seek to mitigate errors rather than completely eliminate them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and describes the hardware demonstrations achieved to date. Commonalities and limitations among the methods are identified, while mention is made of how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified, and the prospects for realizing mitigation-based …