I believe it's true that for any connected smooth manifold M and Lie group G, the "moduli space of flat G-bundles" over M is
where pi_1(M) is the fundamental group of M, and G acts on the homomorphisms from pi_1(M) to G by conjugation.
However, when I seek references for this fact, I'm always led to Narisimhan and Seshadri's paper where they prove this when M is a Riemann surface (and maybe in the analytic or algebraic rather than smooth context).
So, is my belief true? And, almost more importantly: what's a good reference???