We describe several models of Hamiltonian systems in Celestial Mechanics that exhibit dynamical instability.In the cases we consider the instability is owed to the intertwining of the gravitational fields of the Earth,Moon,Sun,planets,etc.Unstable orbits are particularly useful in space mission design,where they can be utilized to produce spacecraft trajectories that travel large distances within the solar system and use as little fuel as possible.In order to exploit such orbits efficiently,one has to understand the geometric structures that organize the dynamics: periodic and quasi-periodic orbits(invariant tori),their stable and unstable manifolds,and the homoclinic and heteroclinic connections between such objects.We describe these geometric structures in some models for the motion of a satellite about the Earth with a realistic potential and for the motion of a satellite in the Earth-Moon,Sun-Earth,and Sun-Jupiter systems.We provide some recipes on how to construct,in each of these models,unstable orbits with prescribed itineraries.In addition,we describe a method,called the `weak stability boundary that was used in the design of the trajectory of the recent NASAs mission GRAIL(2012-2013).We explain the relationship between weak stability boundaries and the stable/unstable manifolds of certain periodic orbits.