  We prove that all hierarchically hyperbolic spaces have finite asymptotic
dimension and obtain strong bounds on these dimensions. One application of this
result is to obtain the sharpest known bound on the asymptotic dimension of the
mapping class group of a finite type surface: improving the bound from
exponential to at most quadratic in the complexity of the surface. We also
apply the main result to various other hierarchically hyperbolic groups and
spaces. We also prove a small-cancellation result namely: if $G$ is a
hierarchically hyperbolic group, $H\leq G$ is a suitable hyperbolically
embedded subgroup, and $N\triangleleft H$ is "sufficiently deep" in $H$, then
$G/\langle\langle N\rangle\rangle$ is a relatively hierarchically hyperbolic
group. This new class provides many new examples to which our asymptotic
dimension bounds apply. Along the way, we prove new results about the structure
of HHSs, for example: the associated hyperbolic spaces are always obtained, up
to quasi-isometry, by coning off canonical coarse product regions in the
original space (generalizing a relation established by Masur--Minsky between
the complex of curves of a surface and Teichm\"{u}ller space).
