Work of Masur and Minsky described properties of the large scale geometry of mapping class groups of surfaces in terms of projections to a family of curve graphs, which they showed to be Gromov hyperbolic. This work inspired the definition by Behrstock, Hagen and Sisto of hierarchically hyperbolic spaces, which have an analogous structure. I will present a result showing that many graphs whose vertices represent multicurves in a surface are hierarchically hyperbolic.