The classical Torelli theorem asserts that a smooth projective curve is determined by its Jacobian together with the principal polar- ization induced by the theta divisor. In modular terms, it asserts that the natural (Torelli) map from the moduli space of smooth projective curves of genus $g$ into the moduli space of principally polarized abelian varieties of dimension $g$ is injective on geometric points. Quite recently, Alexeev has extended the Torelli map, in a geometrically meaningfull way, to modular compactifications of the above moduli spaces, namely the moduli space of Deligne-Mumford stable curves and the moduli space of principally polarized stable semi-abelic pairs. I will report on a joint work with L. Caporaso, in which we study the geometric fibers of the above compactified Torelli map. If time permits, I will also outline some interesting connections with tropical geometry.