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Abstract: In this talk I'll discuss polygonal outer billiards, a dynamical system that is similar in some ways to ordinary billiards except that it is played on the outside of a convex polygon in the plane. For the most part I'll explain my solution to the Moser-Neumann problem, which asks if an outer billiards system can have unbounded orbits. (Yes, for kite-shaped quadrilaterals.) I'll demonstrate the ideas in the proof using software I developed to study this problem. This is work I did around 15 years ago, but it is still my favorite thing to talk about. Towards the end of the talk, I'll try to indicate some of the more recent work I've done, which amounts to building a coarse combinatorial model for the unbounded outer billiards orbits I found.