  We obtain estimates for large and moderate deviations for the capacity of the
range of a random walk on $\mathbb{Z}^d$, in dimension $d\ge 5$, both in the
upward and downward directions. The results are analogous to those we obtained
for the volume of the range in two companion papers [AS17, AS19].
Interestingly, the main steps of the strategy we developed for the latter apply
in this seemingly different setting, yet the details of the analysis are
different
