Explore the topological structure of invariant manifolds in the Circular Restricted 3 Body Problem (CR3BP) and its applications in space mission design and dynamical astronomy. Delve into recent research on missions to Solar System moons, as well as the capture and impact of comets and Near Earth Asteroids. Discover the surprising asymmetry in landing on moons due to nonlinear 3 body effects, particularly the challenges of polar landings despite their scientific importance. Examine how invariant manifolds play a crucial role in understanding the capture and impact of celestial bodies, using examples such as Comet Shoemaker-Levy 9's collision with Jupiter and the temporary capture of Earth's first minimoon, Asteroid 2006 RH120. This one-hour lecture by Martin Lo, presented for the Applied Algebraic Topology Network, builds upon Charles Conley's seminal work from the 1960s and showcases its continued relevance in modern space exploration and astronomical research.