Heatwaves constitute a major threat to human health and ecosystems. Projected increases in heatwave frequency and severity thus lead to the need for prediction to enhance preparedness and minimize adverse impacts. In this Review, we document current capabilities for heatwave prediction at daily to decadal timescales and outline projected changes under anthropogenic warming. Various local and remote drivers and feedbacks influence heatwave development. On daily timescales, extratropical atmospheric blocking and global land–atmosphere coupling are most pertinent, and on subseasonal to seasonal timescales, soil moisture and ocean surface anomalies contribute. Knowledge of these drivers allows heatwaves to be skilfully predicted at daily to weekly lead times. Predictions are challenging beyond timescales of a few weeks, but tendencies for above-average temperatures can be estimated. Further into the future, heatwaves are anticipated to become more frequent, persistent and intense in nearly all inhabited regions, with trends amplified by soil drying in some areas, especially the mid-latitudes. There is also an increased occurrence of humid heatwaves, especially in southern Asia. A better understanding of the relevant drivers and their model representation, including atmospheric dynamics, atmospheric and soil moisture, and surface cover should be prioritized to improve heatwave prediction and projection.