Given a compact Riemann surface, nonabelian Hodge theory relates topological and algebro-geometric objects associated to it. Namely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces associated with the Riemann surface: the character variety (parameterizing representations of the fundamental group) and the Hitchin moduli space (parameterizing Higgs bundles). The P=W conjecture (now a theorem a few times over), predicts that the Hodge theory of the character variety is determined by the topology of the Hitchin space, imposing surprising constraints on each side.   In this talk, I will discuss this conjecture and various things we do and don't understand about it.