Mirror Symmetry, Hitchin's Equations, and Langlands Duality

Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding these statements is extremely difficult without picking a complex structure on $C$ and using Hitchin's equations. We sketch the essential statements both for the ``unramified'' case that $C$ is a compact oriented two-manifold without boundary, and the ``ramified'' case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.