The Langlands conjecture originated as a highly non-trivial generalization of the reciprocity laws in number theory. In my talk, I explain how after certain `geometrization', it becomes a statement about sets (`moduli spaces') of vector bundles on a Riemann surface. The result is a kind of Fourier transform relating sets of vector bundles and local systems on a Riemann surface.

This `geometric Langlands transform' can be used to motivate theorems and conjectures in such diverse areas of mathematics (and physics) as theory of Painleve equations, representation theory of loop groups, autoduality of Jacobians, and mirror symmetry. Some of the relations will be explained in the talk.