A critical question for climate mitigation and adaptation is to understand when and where the signal of changes to climate extremes have persistently emerged or will emerge from the background noise of climate variability. Here we show observational evidence that such persistent changes to temperature extremes have already occurred over large parts of the Earth. We further show that climate models forced with natural and anthropogenic historical forcings underestimate these changes. In particular, persistent changes have emerged in observations earlier and over a larger spatial extent than predicted by models. The delayed emergence in the models is linked to a combination of simulated change ('signal') that is weaker than observed, and simulated variability ('noise') that is greater than observed. Over regions where persistent changes had not occurred by the year 2000, we find that most of the observed signal-to-noise ratios lie within the 16-84% range of those simulated. Examination of simulations with and without anthropogenic forcings provides evidence that the observed changes are more likely to be anthropogenic than nature in origin. Our findings suggest that further changes to temperature extremes over parts of the Earth are likely to occur earlier than projected by the current climate models.