(Joint work with Eran Makover and David Webb) The Jacobian of a Riemann surface is a complex torus with a Kahler structure. Does the Laplace spectrum of a Riemann surface determine the structure of the associated Jacobian? Most known examples of isospectral Riemann surfaces are constructed by a technique of T. Sunada. After reviewing the definition of the Jacobians and the Sunada technique, we give a geometric proof that the Jacobians of Suanda isospectral Riemann surfaces are isogenous; i.e. they have a common finite cover as complex tori. (This result was obtained earlier by Prasad and Rajan.) We also compare the structures of the Jacobians as Kahler manifolds.