After recalling basic facts about the differential geometry of complex vector bundles over a Riemann surface, I will explain the cornerstones of the theory of Higgs bundles (Kobayashi-Hitchin correspondence, non-abelian Hodge theory). I will introduce certain singular solutions (called limiting configurations in joint work with Mazzeo, Swoboda and Witt) to Hitchin's equations, partially compactifying the ends of these moduli spaces.