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.