Let S_g denote the closed, genus g surface. In this talk, we'll discuss the space of flat circle bundles over S_g, also known as the "representation space" Hom(pi_1(S_g), Homeo+(S^1)). The Milnor-Wood inequality gives a lower bound on the number of components of this space (4g-3), but until very recently it was not known whether this bound was sharp. In fact, we still don't know whether the space has finitely many or infinitely many components (!) I'll report on recent work and new tools to understand Hom(\pi_1(S_g), Homeo+(S^1)). In particular, I use dynamical methods to give a new lower bound on the number of components, and find surprising rigidity phenomena for certain geometric representations.