In algebraic geometry, families of curves are studied using moduli spaces. Besides providing deep insights about curves themselves, moduli spaces exhibit important behaviors which help shape our understanding of how geometric objects may be parametrized, giving valuable insight into the development of higher dimensional theory. Geometric aspects of moduli spaces of curves are reflected in their underlying combinatorial structures.

In these lectures, I will discuss combinatorial aspects of the moduli space of curves. In the first talk, I will give an introduction and overview, aimed at those with no prior background about moduli spaces or the moduli space of curves. In the second talk I will discuss recent developments and open problems in the area.