Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around recent work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. This is joint work with Mark Hagen and Alessandro Sisto.