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Classical and relativistic n-body problem: from Levi-Civita to the most advanced interplanetary missions

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.