Quantum-Hall physics and three dimensions

The discovery of the quantum Hall effect (QHE) in 1980 marked a turning point in condensed matter physics: given appropriate experimental conditions, the Hall conductivity σ_xy of a two-dimensional (2D) electron system is exactly quantized. But what happens to the QHE in three dimensions (3D)? Experiments over the past 40 years showed that some of the remarkable physics of the QHE, in particular plateau-like Hall conductivities σ_xy accompanied by minima in the longitudinal resistivity \r{ho}_xx, can also be found in 3D materials. However, since typically \r{ho}_xx remains finite and a quantitative relation between σ_xy and the conductance quantum e^2/h could not be established, the role of quantum Hall physics in 3D remains unsettled. Following a recent series of exciting experiments, the QHE in 3D has now returned to the centre stage. Here, we summarize the leap in understanding of 3D matter in magnetic fields emerging from these experiments.