moduli spaces of curves are highly sophisticated geometric

objects of fundamental importance. Most of the success we have had in

understanding (aspects of) their geometry stems from the fact that

they have a rich combinatorial structure. First off, they are

stratified spaces, where the strata are themselves built out of other

moduli spaces of curves. Secondly, there is a discrete infinite family

of moduli spaces that are connected to each other via tautological

morphisms.

In this talk I will describe in detail this combinatorial structure,

and present some results and some open questions that I hope will

intrigue a crowd of combinatorialists.