A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones $C_m$, called Kunz cones, for which each numerical semigroup with smallest positive element $m$ corresponds to an integer point in $C_m$. It has been shown that if two numerical semigroups correspond to points in the same face of $C_m$, they share many important properties. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.  

In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups. No familiarity with numerical semigroups or polyhedral geometry will be assumed for this talk.