Historically, the $n$-body problem of celestial mechanics gave rise to algebraic topology and to symplectic geometry. After a historical overview I shall present some recent applications of topological and geometrical techniques to open problems of celestial mechanics. This will include ongoing joint work with Jacques Fejoz (Paris-Dauphine) and Richard Montgomery (UC Santa Cruz). We analyze scattering of $n$ celestial bodies by comparing their deterministic motion on $6n$-dimensional phase space with the non-deterministic process of free motion of a billiard ball, including scattering by linear subspaces. Every generic scattering of the billiard is approximated by a scattering process of celestial bodies.