Abstract: In 1982 Simon Donaldson proved that flat unitary connections (on Riemann surfaces) correspond to stable holomorphic vector bundles. This is a specific case of a general phenomenon: stable objects in algebraic geometry correspond to extremal objects in differential geometry. One might therefore ask what stable algebraic objects correspond to flat connections which are not necessarily unitary. In 1987 Nigel Hitchin introduced Higgs bundles as the answer to this question. In this talk I will define Higgs bundles and discuss these correspondences, and time permitting will discuss some interesting applications of the theory: integrable systems, Mirror symmetry, the Langlands and geometric Langlands programs, non-Abelian cohomology, and theoretical physics.