Navigation requires a sense of direction ('compass'), which in mammals is thought to be provided by head-direction cells, neurons that discharge when the animal's head points to a specific azimuth. However, it remains unclear whether a three-dimensional (3D) compass exists in the brain. Here we conducted neural recordings in bats, mammals well-adapted to 3D spatial behaviours, and found head-direction cells tuned to azimuth, pitch or roll, or to conjunctive combinations of 3D angles, in both crawling and flying bats. Head-direction cells were organized along a functional-anatomical gradient in the presubiculum, transitioning from 2D to 3D representations. In inverted bats, the azimuth-tuning of neurons shifted by 180 degrees, suggesting that 3D head direction is represented in azimuth x pitch toroidal coordinates. Consistent with our toroidal model, pitch-cell tuning was unimodal, circular, and continuous within the available 360 degrees of pitch. Taken together, these results demonstrate a 3D head-direction mechanism in mammals, which could support navigation in 3D space.