Abstract: The Langlands Program was launched in the 60s with the goal of relating Galois representations and automorphic forms. In recent years the geometric Langlands Program has been developed which points to a mysterious dualitiy between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. It turns out that this duality is closely related to the Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two dual Lie groups. I will explain this connection and talk about my recent work with Edward Witten in which we use it to gain insights into the phenomenon of "endoscopy" in the geometric Langlands Program.