The mysterious geometry of Artin groups

Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between Coxeter groups and Artin groups as well as the associated topological spaces used to investigate them. The second part describes the location of the border between the Artin groups we understand at a very basic level and those that remain fundamentally mysterious. The third part highlights those collections of Artin groups (and their relatives) that are not currently understood but which we are likely to understand sometime soon.