In the study of compact Riemann surfaces, the Jacobian variety plays an important role. To each Riemann surface, S, one can associate to S a topological genus g. The Jacobian variety is the quotient space \mathbb{C}^g/\Lambda, where
\Lambda is the \mathbb{Z} linear span of the periods of S. In the talk, we shall see more precisely what this means, and how the Jacobian arises naturally.

The Torelli theorem states that given a Jacobian variety, as well as an additional piece of data called the principal polarisation, the compact Rieman surface is determined up to isomorphism. This assures us that in studying a Riemann surface using via its Jacobian variety, no information is lost. In the talk, we will see a few highlights of its proof, which uses techniques in algebraic geometry.