The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space M$_g,_n$. I will give results on the topology of tropical M$_1,_n$, joint with Galatius and Payne, and on tropical M$_2,_n$, obtaining as corollaries new calculations on the top-weight cohomology of the complex moduli spaces M$_1,_n$ and M$_2,_n$.