Dynamical systems in low dimensions, such as interval exchanges or flows on surfaces, come in natural families and their moduli spaces are objects of intrinsic interest. Riemann surfaces and their geometry play a key role, and after introducing the basics I will give an overview of recent results in this area. I will then discuss extensions of these concepts to higher dimensions, where the geometry of K3 surfaces comes in. Generalizing counts of closed billiard trajectories in rational-angled polygons, I will explain how to count a higher-dimensional analogue on K3s. The necessary background will be provided.