The aim of this seminar is to highlight the combinatorial aspects that arise from studying the compactifications of the moduli space of smooth curves. We will provide an overview of the construction of the tropical moduli space of curves, which can be developed in an entirely combinatorial manner. We will then explore how this relates to the classical Deligne-Mumford compactification, passing through Berkovich analytic spaces. If time permits, we will also introduce the notion of cone stacks, which serve as a tropical analogue to the Deligne-Mumford stacks.