Studying the spatial extent of heat extremes is essential to understand their impacts and physical mechanisms. Although some studies have examined the structure and extent of heat extremes at the surface, few have quantified their horizontal and vertical spatial extent and analyzed their variations with characteristics of heatwaves such as intensity and location. Here, we propose a global and systematic description of the three-dimensional structure of temperature anomalies during surface heat extremes using ERA5 reanalysis data. This is achieved by performing a percentile-based contour-stacking algorithm on the daily mean temperature anomaly field to obtain a representation of the three-dimensional structure of temperature anomalies in the troposphere. We characterize this structure in terms of area, intensity and geometry. We find that near-surface heat extremes have a relatively uniform average area globally, corresponding to the area covered by a circle with a radius of 800 km, although with important geographical variations. The intensity and area of temperature anomalies decrease with height, but much faster in the tropics than in the extratropics. The shape of temperature anomalies at all levels is not circular and is better described by an ellipse with an axis ratio of 0.5–0.6. These ellipses have a systematic preferential orientation that depends mostly on latitude and topography. Tropical temperature anomalies are zonally oriented, but the orientation leans poleward towards higher latitudes, following the direction of anticyclonic wave breaking. Although there is a robust positive correlation between the intensity and the spatial extent of heat extremes in the mid-troposphere, this relationship is more complex and weaker at the surface. We finally advocate against the image of a "heat dome" to describe the vertical structure of temperature anomalies associated to surface heat extremes.