The Brauer group of an algebraic variety X is the group of Azumaya algebras over X, or equivalently the group of Severi-Brauer varieties over X. It is a central object in algebraic and arithmetic geometry, being for example one of the first ways to produce counterexamples to Noether's problem of whether, given a representation V of a finite group G, the quotient V/G is rational. 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. Cohomological invariants are a classical theory of invariants of algebraic groups, providing an arithmetic equivalent to characteristic classes. In my PhD thesis, I extended the concept to a theory of invariants for general algebraic stacks, and computed them for the moduli stacks of elliptic and hyperelliptic curves. I will talk about some recent results, joint with A. Di Lorenzo, where we show that cohomological invariants can be used to compute the Brauer groups of moduli stacks, and use them to 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.