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The discovery of the quantum Hall effect in 1980 marked a turning point in condensed matter physics. The measurement of the Hall resistance showed that electronic resistance could be defined precisely in terms of fundamental constants, even in a disordered and irregular sample. Over the past 40 years, this discovery has inspired new theories and led to experimental discoveries in a range of fields including topology, magnetism and 2D systems. This Collection of Review, News and Commentary articles from various Nature Research journals celebrates the diverse legacy of this discovery.