Tropical Teichmüller space and the moduli space of curves

I will report on joint work with Margarida Melo and Filippo Viviani exhibiting the moduli space of tropical curves as a quotient of tropical Teichm\"uller space, a closure of Culler-Vogtmann Outer Space, by the action of the outer automorphisms of the free group.  We also provide a lift of the tropical Torelli map to a tropical period map from tropical Teichm\"uller space to the space of positive semidefinite forms with rational nullspace.