  This paper is a combinatorial and computational study of the moduli space of
tropical curves of genus g, the moduli space of principally polarized tropical
abelian varieties, and the tropical Torelli map. These objects were introduced
recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the
category of stacky fans, of which the aforementioned moduli spaces are objects
and the Torelli map is a morphism. We compute the poset of cells of tropical
M_g and of the tropical Schottky locus for genus at most 5. We show that
tropical A_g is Hausdorff, and we also construct a finite-index cover for A_3
which satisfies a tropical-type balancing condition. Many different
combinatorial objects, including regular matroids, positive semidefinite forms,
and metric graphs, play a role.
