Explore the spatial restricted three-body problem in this advanced mathematics lecture. Delve into Poincaré's approach to finding closed orbits and its generalization to the spatial case. Learn about global hypersurfaces of section, Hamiltonian return maps, and a higher-dimensional version of the Poincaré-Birkhoff theorem. Examine the construction associating Reeb dynamics on a moduli space of holomorphic curves to the given dynamics. Cover topics such as the integrable limit cases, Moser regularization, contact geometry, open book decompositions, and fixed point theory of Hamiltonian twist maps. Gain insights into the physical interpretation of spatial vs planar orbits and the concept of holomorphic shadows in this joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry seminar presented by Agustin Moreno from Uppsala University.