Hodge theory studies the fine structure of the cohomology of complex projective varieties (or compact Kähler manifolds) and the information they contain. In the first part of the talk, we will explore basic concepts of Hodge theory, such as Hodge decomposition, polarized Hodge structures, and period domains, with several examples. In the second part, we prove Torelli theorem for curves, given by Andreotti. It says that the curve Jacobian as a polarized variety uniquely defines the curve. For this proof we recall the standard notions of the Abel-Jacobi map, the Gauss map and Theta divisors.