  Quantum systems with SU($N$) symmetry are paradigmatic settings for quantum
many-body physics. They have been studied for the insights they provide into
complex materials and their ability to stabilize exotic ground states.
Ultracold alkaline-earth atoms were predicted to exhibit SU($N$) symmetry for
$N=2I+1=1,2,\ldots,10$, where $I$ is the nuclear spin. Subsequent experiments
have revealed rich many-body physics. However, alkaline-earth atoms realize
this symmetry only for fermions with repulsive interactions. In this paper, we
predict that ultracold molecules shielded from destructive collisions with
static electric fields or microwaves exhibit SU($N$) symmetry, which holds
because deviations of the s-wave scattering length from the spin-free values
are only about 3\% for CaF with static-field shielding and are estimated to be
even smaller for bialkali molecules. They open the door to $N$ as large as $32$
for bosons and $36$ for fermions. They offer important features unachievable
with atoms, including bosonic systems and attractive interactions.
