I will report on a joint work in progress with M. Chan and M. Melo in which we study a tropical analogous of the classical Teichmuller theory. In particular, we prove that the moduli space of pure tropical curves (constructed in a joint work with S. Brannetti and M. Melo) is the quotient of the outer space (constructed by Culler-Vogtmann) by the outer automorphism group of the free group. Next we lift the tropical Torelli map to a map between the outer space and the cone of positive definite quadratic forms. Finally, by using results in joint works with L. Caporaso and with S. Brannetti and M. Melo, we study the fibers and the image of this lifted Torelli map.