Hierarchically hyperbolic groups form a vast family of groups

satisfying a coarse but highly organised form of non-positive curvature. Notable examples include hyperbolic groups, mapping class groups, compact special groups and many Artin groups. In this talk, I will discuss recent joint work with Mark Hagen in which we investigated the behaviour of their virtually abelian subgroups, proving in particular a coarse analogue of the CAT(0) flat torus theorem and flat closing problem for this class of groups.