The ABC conjecture, first posed in the 1980s, is one of the central problems in number theory. Its statement is completely elementary, while at the same time it is closely related to numerous conjectures and deep theorems that were already known. In 2012 Shinichi Mochizuki announced a proof of the ABC conjecture, but the path to deciding if the proof is valid has been problematic to say the least. We will discuss what the ABC conjecture says, some of its consequences and alternative formulations, and what is going on with Mochizuki's work.