The moduli spaces of stable vector bundles over a Riemann surface have been intensively studied from various points of view (algebraic geometry, topology or gauge theory to name a few). Here I want to present another viewpoint pioneered by Hitchin, namely the symplectic geometry of their cotangent bundles. Based on this I will report on some recent work joint with Mazzeo, Swoboda and Weiß concerning the compactification of these spaces.