  Geometric Langlands duality is usually formulated as a statement about
Riemann surfaces, but it can be naturally understood as a consequence of
electric-magnetic duality of four-dimensional gauge theory. This duality in
turn is naturally understood as a consequence of the existence of a certain
exotic supersymmetric conformal field theory in six dimensions. The same
six-dimensional theory also gives a useful framework for understanding some
recent mathematical results involving a counterpart of geometric Langlands
duality for complex surfaces. (This article is based on a lecture at the Raoul
Bott celebration, Montreal, June 2008.)
