  Towards the end of nineteenth century, Celestial Mechanics provided the most
powerful tools to test Newtonian gravity in the solar system, and led also to
the discovery of chaos in modern science. Nowadays, in light of general
relativity, Celestial Mechanics leads to a new perspective on the motion of
satellites and planets. The reader is here introduced to the modern formulation
of the problem of motion, following what the leaders in the field have been
teaching since the nineties. In particular, the use of a global chart for the
overall dynamics of N bodies and N local charts describing the internal
dynamics of each body. The next logical step studies in detail how to split the
N-body problem into two sub-problems concerning the internal and external
dynamics, how to achieve the effacement properties that would allow a
decoupling of the two sub-problems, how to define external-potential-effacing
coordinates and how to generalize the Newtonian multipole and tidal moments.
The review paper ends with an assessment of the nonlocal equations of motion
obtained within such a framework, a description of the modifications induced by
general relativity of the theoretical analysis of the Newtonian three-body
problem, and a mention of the potentialities of the analysis of solar-system
metric data carried out with the Planetary Ephemeris Program.
