This thesis is devoted to the study of the symplectic algebraic geometry of the moduli spaces of the nonabelian Hodge correspondence: namely, we consider character varieties of (open) Riemann surfaces and the moduli spaces of semistable Higgs bundles on a closed Riemann surface and aim at giving an answer to the following two questions: are those moduli spaces symplectic singularities? Do they admit symplectic resolutions? In the case of Higgs bundles on closed Riemann surfaces, we are able to completely solve the problem, using the work of Bellamy and Schedler, in combination with Simpson’s Isosingularity theorem. For what concerns the Betti moduli spaces, we take a different perspective and realise them as (singular open subsets of) multiplicative quiver varieties, and we study the aforementioned problem for the latter varieties. The present work represents a first step of a series of investigations on the symplectic algebraic geometry involved in the nonabelian Hodge correspondence.