While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields.
I will report on recent works with A. Di Lorenzo. Using the theory of cohomological invariants we completely describe the Brauer group of the moduli stacks of hyperelliptic curves over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It is generated by elements coming from the base field, cyclic algebras, an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. There are two natural compactifications, the first obtained by taking stable hyperelliptic curves and the second by taking admissible covers. It turns out that the Brauer group of the first is trivial, while for the second it is almost as large as in the non-compact case.