  We compute the $\ell$-primary torsion of the Brauer group of the moduli stack
of smooth curves of genus three over any field of characteristic different from
two. We achieve this result by computing the cohomological invariants of this
stack. Along the way, we also compute the cohomological invariants and the
Brauer group of the moduli stacks of smooth plane curves of degree $d$ over any
algebraically closed field of characteristic different from two and coprime to
$d$.
