What are the predictions of climate models, should we believe them, and are they falsifiable? Probably the most iconic and influential result arising from climate models is the prediction that, dependent on the rate of increase of CO2 emissions, global and annual mean temperature will rise by around 2–4°C over the 21st century. We argue that this result is indeed credible, as are the supplementary predictions that the land will on average warm by around 50% more than the oceans, high latitudes more than the tropics, and that the hydrological cycle will generally intensify. Beyond these and similar broad statements, however, we presently find little evidence of trustworthy predictions at fine spatial scale and annual to decadal timescale from climate models.For further resources related to this article, please visit the WIREs website.Conflict of interest: The authors have declared no conflicts of interest for this article.