The notion of a connection on a vector bundle is of tremendous importance both in differential geometry as well as in mathematical physics. Using connections one can understand the local geometry at points of a vector bundle and differentiate sections along vector fields. Moreover, connections give rise to fundamental geometric invariants such as the curvature. Physically, connections are called gauge fields or vector fields, and their curvature is the corresponding field strength. Gauge theories provide the appropriate mathematical framework to describe forces in Physics. For example, the relativistic vector potential in Maxwell’s theory is a connection, of which the electromagnetic tensor describing the electric and magnetic fields is the curvature. The Riemann–Hilbert correspondence relates the geometry of flat vector bundles to the topological structure of the base manifold. In this chapter, we briefly review the theory of vector bundles with connections over a Riemann surface; an abundance of texts can be found in the literature for a proper introduction in differential geometry of complex vector bundles, for instance Kobayashi and Nomizu (Foundations of differential geometry. I, 1963) or Wells (Differential analysis on complex manifolds. With a new appendix by Oscar Garcia-Prada, 2008). For readers with more physical background, we warmly recommend (Baez and Muniain, Gauge fields, knots and gravity, 1995).