  We place the representation variety in the broader context of abelian and
nonabelian cohomology. We outline the equivalent constructions of the moduli
spaces of flat bundles, of smooth integrable connections, and of holomorphic
integrable connections over a compact Kaehler manifold. In addition, we
describe the moduli space of Higgs bundles and how it relates to the
representation variety. We attempt to avoid abstraction, but strive to present
and clarify the unifying ideas underlying the theory.
