We describe how to determine transfers between libration point orbits and either the surface of the Moon or a Low Earth Orbit within the Circular Restricted Three – Body Problem (CR3BP) assumptions. Moreover, we explain how to refine such trajectories to ones verifing more comprehensive equations of motion. We are interested in seeing how the geometry of the nominal target orbits and of the associated stable manifolds drives the connections and also how much reliable the CR3BP is. The main tools we take advantage of are the Lindstedt–Poincaré semi-analytical method, differential correction procedures and an optimizer.