I will give a hopefully accessible introduction to some work on tropical moduli spaces of curves and abelian varieties. I will report on joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe, in which we find new rational cohomology classes in the moduli space $A_{g}$ of abelian varieties using tropical techniques. And I will try to touch on a new point of view on this topic, namely that of differential forms on tropical moduli spaces, following the work of Francis Brown.