In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a factor system, and the role of the curve graph is played by the contact graph. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur-Minsky-style distance formula.We then define a hierarchically hyperbolic space; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichmüller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock-Minsky, Eskin-Masur-Rafi, Hamenstädt, and Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.

We introduce a number of tools for finding and studying hierarchically hyperbolic spaces (HHS), a rich class of spaces including mapping class groups of surfaces, Teichmüller space with either the Teichmüller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHS satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hierarchy machinery. We then study examples of HHS; for instance, we prove that when M is a closed irreducible 3-manifold then π1M is an HHS if and only if it is neither N il nor Sol. We establish this by proving a general combination theorem for trees of HHS (and graphs of HH groups). We also introduce a notion of "hierarchical quasiconvexity", which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.