To a Riemann Surface S of genus g (compact complex manifold of complex dimension 1) one can associate a g-dimensional complex torus J(S) called the Jacobian of S. Kodaira's embedding theorem states that a compact complex manifold M can be holomorphically embedded into a complex projective space iff there exists a Kahler form w on M whose cohomology class is integral. It turns out that there is a natural way to choose such a Kahler form on the Jacobian of a Riemann surface which proves that J(S) can be embedded into P(n) for some n. Torelli's theorem states that two Riemann surfaces S and R are isomorphic as complex manifolds iff there exists a biholomorphism between J(S) and J(R) taking the corresponding Kahler classes to each other. In this talk I will explain how to construct J(S) and its integral Kahler class w.