Metal halide perovskite materials are an emerging class of solution-processable semiconductors with considerable potential for use in optoelectronic devices1-3. For example, light-emitting diodes (LEDs) based on these materials could see application in flat-panel displays and solid-state lighting, owing to their potential to be made at low cost via facile solution processing, and could provide tunable colours and narrow emission line widths at high photoluminescence quantum yields4-8. However, the highest reported external quantum efficiencies of green- and red-light-emitting perovskite LEDs are around 14 per cent7,9 and 12 per cent8, respectively—still well behind the performance of organic LEDs10-12 and inorganic quantum dot LEDs13. Here we describe visible-light-emitting perovskite LEDs that surpass the quantum efficiency milestone of 20 per cent. This achievement stems from a new strategy for managing the compositional distribution in the device—an approach that simultaneously provides high luminescence and balanced charge injection. Specifically, we mixed a presynthesized CsPbBr3 perovskite with a MABr additive (where MA is CH3NH3), the differing solubilities of which yield sequential crystallization into a CsPbBr3/MABr quasi-core/shell structure. The MABr shell passivates the nonradiative defects that would otherwise be present in CsPbBr3 crystals, boosting the photoluminescence quantum efficiency, while the MABr capping layer enables balanced charge injection. The resulting 20.3 per cent external quantum efficiency represents a substantial step towards the practical application of perovskite LEDs in lighting and display.