Participation in permissionless blockchains results in competition over system resources, which needs to be controlled with fees. Until recently, Ethereum's fee mechanism was implemented via a first-price auction that resulted in unpredictable fees as well as other inefficiencies. Launched on August 5, 2021, EIP-1559 is an improved proposal that introduces a number of innovative features such as a dynamically adaptive basefee that is burnt, instead of being paid to the miners. Despite intense interest in understanding its properties, several basic questions such as whether and under what conditions does this protocol self-stabilize have remained elusive thus far.