A hierarchically hyperbolic structure gives a way of reducing the study of a given metric space to the study of a specified family of hyperbolic spaces. Spaces with hierarchically hyperbolic structures include CAT(0) cube complexes admitting a proper and cocompact isometric action, mapping class groups, Teichmuller spaces with either the Teichmuller or the Weil-Petersson metric and many 3-manifold groups.

I will outline what a hierarchically hyperbolic structure is and how to give one to a CAT(0) cube complex admitting a factor system, which is a "large enough" locally finite collection of convex subcomplexes. Finally, I will give applications, in particular one regarding acylindrical actions.
Based on joint works with J. Behrstock and M. Hagen