The rechargeable lithium–air battery has the highest theoretical specific energy of any rechargeable battery and could transform energy storage if a practical device could be realized. At the fundamental level, little was known about the reactions and processes that take place in the battery, representing a significant barrier to progress. Here, we review recent advances in understanding the chemistry and electrochemistry that govern the operation of the lithium–air battery, especially the reactions at the cathode. The mechanisms of O2 reduction to Li2O2 on discharge and the reverse process on charge are discussed in detail, as are their consequences for the rate and capacity of the battery. The various parasitic reactions involving the cathode and electrolyte during discharge and charge are also considered. We also provide views on understanding the stability of the cathode and electrolyte and examine design principles for better lithium–air batteries.