In moduli theory, we want to define spaces parametrizing objects up to certain notions of isomorphism. In this talk we will discuss a fundamental way of constructing such moduli: geometric invariant theory, i.e., Mumford’s formalism to define quotients under group actions in algebraic geometry. Finally, we will relate the cohomology of GIT quotients with equivariant cohomology via quotient stacks, HKKN stratifications and quantization.