The talk will be a general account of Mirror Symmetry (from a purely mathematical point of view) and its relations to the geometric Langlands conjecture. I will concentrate on some simple cases related to a particularly simple class of hyper-Kähler varieties, namely moduli spaces of flat bundles on curves. In this case, Mirror Symmetry predicts various conjectures, e.g., on invariants of the moduli spaces of flat bundles. I will talk about some partial results, using various approaches ranging from non-abelian Hodge theory to calculations involving counting of points of varieties defined over finite fields.