The last decade has produced record-breaking heat waves in many parts of the world. At the same time, it was globally the warmest since sufficient measurements started in the 19th century. Here we show that, worldwide, the number of local record-breaking monthly temperature extremes is now on average five times larger than expected in a climate with no long-term warming. This implies that on average there is an 80 % chance that a new monthly heat record is due to climatic change. Large regional differences exist in the number of observed records. Summertime records, which are associated with prolonged heat waves, increased by more than a factor of ten in some continental regions including parts of Europe, Africa, southern Asia and Amazonia. Overall, these high record numbers are quantitatively consistent with those expected for the observed climatic warming trend with added stationary white noise. In addition, we find that the observed records cluster both in space and in time. Strong El Niño years see additional records superimposed on the expected long-term rise. Under a medium global warming scenario, by the 2040s we predict the number of monthly heat records globally to be more than 12 times as high as in a climate with no long-term warming.