Let C be a (smooth projective algebraic) curve, in other words, a Riemann surface. It is well known that the Jacobian J of C is a self-dual complex torus, that is, J is identified with the space of topologically trivial line bundles on J. Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is smooth, but no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J. In this talk, I consider curves C with double singular points. The main result is that J' is self-dual: it is identified with a space of torsion-free sheaves on J'. I also plan to discuss the Fourier-Mukai transform arising from the autoduality. The compactified Jacobians play a role in the geometric Langlands correspondence (for GL(n)), where they appear as fibers of the Hitchin fibration. The autoduality of compactified Jacobians can thus be viewed as a `classical limit' of the Langlands correspondence.