The goal of this school is to explain several methods and results from modern Riemannian and symplectic geometry and from Hamiltonian systems, and to use them to study old problems in celestial mechanics. Among the tools are symplectic embeddings by elementary and less elementary methods, J-holomorphic curves, and systolic inequalities in Riemannian, contact and symplectic geometry. We shall apply these tools to study very explicit problems on the motion of Moon, Earth, and Sun in the form of the restricted 3-body problem and several of its limits. This will show that the modern methods we use are by no means dry mathematical abstractions, but have their origin in concrete problems in mechanics, geometrical optics, and celestial mechanics.