Artin groups form a class of groups generalising braid groups, and whose geometry remains elusive. In recent years, there has been a lot of progress in understanding hyperbolic features of large classes of Artin groups beyond the right-angled case, in particular highlighting striking similarities with mapping class groups of hyperbolic surfaces. In this talk, I will mention some of these results, and I will talk in particular about current work with M. Hagen and A. Sisto showing that Artin groups of extra-large type are hierarchically hyperbolic. All these results rely in one way or another on the structure of certain complexes that have (sometimes distant) connections with polyhedral products.

This talk will mention joint work with J. Behrstock, A. Genevois, M. Hagen, P. Przytycki, and A. Sisto (in various overlapping subsets).