Effective Computations in Celestial Mechanics and Astrodynamics (CISM Course on ``Modern Methods of Analytical Mechanics and their Applications'', Udine, June 16 to 20, 1997) Carles Sim\'o, Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona Gran Via 585, 08007 Barcelona Problems in Celestial Mechanics and Astrodynamics are considered under the point of view of Hamiltonian Dynamical Systems. A main tool to analyze the dynamics consists in studying the {\em skeleton} of the system, that is, the invariant objects (fixed points, periodic orbits and invariant tori) as well as their related stable, unstable and centre manifolds. Methods to compute these objects are presented. They are analytical, with the symbolic implementacion, purely numerical or a combination of both. Several examples are presented.