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Models disagree on a significant number of responses to climate change, such as climate feedback, regional changes, or the strength of equilibrium climate sensitivity. Emergent constraints aim to reduce these uncertainties by finding links between the inter-model spread in an observable predictor and climate projections. In this paper, the concepts underlying this framework are recalled with an emphasis on the statistical inference used for narrowing uncertainties, and a review of emergent constraints found in the last two decades. Potential links between highlighted predictors are explored, especially those targeting uncertainty reductions in climate sensitivity, cloud feedback, and changes of the hydrological cycle. Yet the disagreement across emergent constraints suggests that the spread in climate sensitivity can not be significantly narrowed. This calls for weighting the realism of emergent constraints by quantifying the level of physical understanding explaining the relationship. This would also permit more efficient model evaluation and better targeted model development. In the context of the upcoming CMIP6 model intercomparison a growing number of new predictors and uncertainty reductions is expected, which call for robust statistical inferences that allow cross-validation of more likely estimates. [ABSTRACT FROM AUTHOR]