Tropicalizing Moduli Spaces and Applications

We describe intrinsic relations between the moduli space of smooth projective curves, its compactification via Deligne-Mumford stable curves, and the moduli space of tropical curves. The key point lies in describing a category of combinatorial objects governing the topological type stratification of both moduli spaces. We also illustrate remarkable applications of this interplay on understanding the geometry of the original space and explain similar applications to other moduli spaces.