Magnesium batteries have long been pursued as potentially low-cost, high-energy and safe alternatives to Li-ion batteries. However, Mg2+ interacts strongly with electrolyte solutions and cathode materials, leading to sluggish ion dissociation and diffusion, and consequently low power output. Here we report a heterogeneous enolization chemistry involving carbonyl reduction (C=O↔C–O−), which bypasses the dissociation and diffusion difficulties, enabling fast and reversible redox processes. This kinetically favoured cathode is coupled with a tailored, weakly coordinating boron cluster-based electrolyte that allows for dendrite-free Mg plating/stripping at a current density of 20 mA cm−2. The combination affords a Mg battery that delivers a specific power of up to 30.4 kW kg−1, nearly two orders of magnitude higher than that of state-of-the-art Mg batteries. The cathode and electrolyte chemistries elucidated here propel the development of magnesium batteries and would accelerate the adoption of this low-cost and safe battery technology.