  We introduce a number of tools for finding and studying \emph{hierarchically
hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups
of surfaces, Teichm\"{u}ller space with either the Teichm\"{u}ller 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 HHSs 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 HHSs; for instance, we prove that when $M$ is a closed
irreducible $3$--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 for
trees of HHSs (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.
