A major challenge in dynamics on moduli spaces is to understand the behavior of the horocycle flow. We will motivate this problem and discuss what is known and what is not known about it, focusing on the genus 2 case. Specific topics to be covered include: * SL_2(R) orbit closures and invariant measures in genus 2. * Quantitative nondivergence. * The structure of minimal sets. * Rel and real-rel, and their interaction with the horocycle flow * Horizontal data diagrams and other invariants for horocycle invariant measures. * Classification of measures and orbit-closures in the eigenform loci. * Recent and not-so-recent examples of unexpected measures and orbit-closures.