  In this expository article we give a categorical definition of the integral
cohomology ring of a stack. We show that for quotient stacks the categorical
cohomology may be identified with equivariant cohomology. Via this
identification we show that for Deligne-Mumford quotient stacks this cohomology
is rationally isomorphic to the rational cohomology of the coarse moduli space.
The theory is presented with a focus on the stacks of smooth and stable curves.
