I intend to give a true introduction, accessible to beginning graduate students, to the topics in the title: what is tropical geometry? What is a moduli space? And how can one be used to study the other?  Then I'll discuss some aspects of joint work with Francis Brown, Søren Galatius, and Sam Payne, in which we identify a Hopf algebraic structure on the weight 0 subspace of the compactly supported cohomology of the moduli space of abelian varieties and deduce a number of consequences.