I will discuss a collection of hyperbolic graphs associated to a CAT(0) cube complex and explain how the geometry of the cube complex can be recovered -- up to quasi-isometry -- from its shadows on these graphs.  I will explain how this mirrors the Masur-Minsky theory enabling the study of the mapping class group of a surface via projections to curve graphs of subsurfaces.  I'll then define "hierarchical hyperbolicity", which is a common generalisation of these two classes of examples, and discuss some applications.  This is based on joint work with J. Behrstock and A. Sisto