The horocycle flow on the unit tangent bundle of a hyperbolic surface has some amazing dynamical properties. Results of Ratner and others show that, more generally, unipotent flows in homogeneous dynamics have analogous amazing properties. The horocycle flow has a natural analog in the world of flat surfaces and it is natural to ask whether this flow shares these amazing properties. In this talk I will describe joint results with Jon Chaika and Barak Weiss in which we show that in important respects this analogy breaks down.