Celestial Mechanics and Dynamical Astronomy (2006) DOI 10.1007/s10569-005-6215-x REVIEW ARTICLE KAM tori for N -body problems: a brief history A. Celletti · L. Chierchia Received: 28 November 2005 / Accepted: 24 December 2005 © Springer Science+Business Media B.V. 2006 Abstract We review analytical (rigorous) results about the existence of invariant tori for planetary many-body problems. Keywords Computer-assisted proofs · Invariant tori · KAM theory · N-body problem · Small divisor problems 1 Introduction In this paper, we review analytical results concerning the existence of KAM tori (smooth invariant tori, for a nearly integrable Hamiltonian system, on which the flow is quasi-periodic with Diophantine frequencies) in the context of the planetary many-body problem. The main body of the paper is divided in two sections and two appendices. In Sect. 2, general existence theorems for the planetary (1 + n)-body problem are discussed. In particular, after a brief reminder about the Hamiltonian setting for the many-body problem (Sect. 2.1) and about classical KAM theory (Sect. 2.2), it is shown how Kolmogorov’s 1954 theorem yields easily the existence of KAM tori in the special non-degenerate case of the restricted, planar, circular three-body problem (Sect. 2.2.2). Kolmogorov’s theorem, on the other hand, does not apply to the general case because of the proper degeneracy of the (1 + n)-body problem, when n  2. In this context, Arnold (1963), stated a general result, which he proved only in the A. Celletti (B ) Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma Italy e-mail: celletti@mat.uniroma2.it L. Chierchia Dipartimento di Matematica, Università “Roma Tre”, Largo S. L. Murialdo 1, I-00146 Roma, Italy e-mail: luigi@mat.uniroma3.it