Abstract: The space of phylogenetic trees on n leaves arises naturally in tropical geometry through the tropical Grassmannian trop Gr(2,n). The space of equidistant trees on n leaves is the tropicalization of M_{0,n}, which is tropically convex. In this talk, I will present recent work using tropical tree spaces for phylogenetic statistics and inference (joint with Curiel, Sabol, Talbut, and Yoshida). I will also discuss a conjectural analogue of the space of equidistant trees for type C (joint with Igor Makhlin).