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We introduce a number of tools for finding and studying \emph{hierarchicallyhyperbolic spaces (HHS)}, a rich class of spaces including mapping class groupsof surfaces, Teichm\"{u}ller space with either the Teichm\"{u}ller orWeil-Petersson metrics, right-angled Artin groups, and the universal cover ofany compact special cube complex. We begin by introducing a streamlined set ofaxioms defining an HHS. We prove that all HHSs satisfy a Masur-Minsky-styledistance formula, thereby obtaining a new proof of the distance formula in themapping class group without relying on the Masur-Minsky hierarchy machinery. Wethen study examples of HHSs; for instance, we prove that when $M$ is a closedirreducible $3$--manifold then $\pi_1M$ is an HHS if and only if it is neither$Nil$ nor $Sol$. We establish this by proving a general combination theorem fortrees of HHSs (and graphs of HH groups). We also introduce a notion of"hierarchical quasiconvexity", which in the study of HHS is analogous to therole played by quasiconvexity in the study of Gromov-hyperbolic spaces.

Hierarchically hyperbolic spaces II: Combination theorems and the distance formula