  Shephard groups are common generalizations of Coxeter groups, Artin groups,
and graph products of cyclic groups. Their definition is similar to that of a
Coxeter group, but generators may have arbitrary order rather than strictly
order 2. We extend a well known result that Coxeter groups are
$\mathrm{CAT}(0)$ to a class of Shephard groups that have "enough" finite
parabolic subgroups. We also show that in this setting, if the associated
Coxeter group is type (FC), then the Shephard group acts properly and
cocompactly on a $\mathrm{CAT}(0)$ cube complex. As part of our proof of the
former result, we introduce a new criteria for a complex made of $A_3$
simplices to be $\mathrm{CAT}(1)$.
