The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic.In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum. The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.