  The n-body problem is one of the most important issue in Celestial Mechanics.
This article aims to retrace the historical and scientific events that led the
Paduan mathematician, Tullio Levi-Civita, to deal with the problem first from a
classic and then a relativistic point of view. We describe Levi-Civita's
contributions to the theory of relativity focusing on his epistolary exchanges
with Einstein, on the problem of secular acceleration and on the proof of
Brillouin's cancellation principle. We also point out that the themes treated
by Levi-Civita are very topical. Specifically, we analyse how the mathematical
formalism used nowadays to test General Relativity can be found in
Levi-Civita's texts and evolves over the years up to the current Parametrized
version of the Post-Newtonian approximation (PPN) which is used in high
precision contexts such as important space missions designed also to test
General Relativity and which aim to estimate with very high accuracy the PPN
parameters.
