Title: Splitting, return maps and confined motion in the planar RTBP: Theory and praxis Author: Carles Sim\'o Dept. Matem\`atica Aplicada i An\`alisi, UB, Barcelona, Catalunya Slides of a talk at the Edinburgh Mathematical Society - Societat Catalana de Matem\`atiques joint meeting Barcelona, 28-30 May 2015 Abstract: Return maps to a given domain close to a broken separatrix are useful to understand many dynamical properties. The talk will concentrate on symplectic 2D maps and related problems. The main ingredients are the return time to the domain and the splitting of separatrices. Different models are obtained, depending on the problem at hand. They allow to produce realistic quantitative estimates on the boundaries of confined motion. The methodology is applied to the planar restricted three-body problem. Comparing the theoretical predictions with a careful numerical study one has deeper understanding and a source of new problems. This is a joint work with Regina Mart\'{\i}nez.