MPEJ Volume 10, No. 3, 37 pp. Received: Feb 17, 2004. Revised: Feb 26, 2004. Accepted: Mar 1, 2004. D. Salamon The Kolmogorov-Arnold-Moser theorem ABSTRACT: This paper gives a self contained proof of the perturbation theorem for invariant tori in Hamiltonian systems by Kolmogorov, Arnold, and Moser with sharp differentiablility hypotheses. The proof follows an idea outlined by Moser in~\cite{M4} and, as byproducts, gives rise to uniqueness and regularity theorems for invariant tori.\footnote {The present paper was written in 1986 while I was a postdoc at ETH Z\"urich. I didn't publish it at the time because the results are well known and the paper is of expository nature. The paper is reproduced here with the following changes: there are a few updates in the introduction, a mistake in Lemma~\ref{le:approx1} and the proof of Theorem~\ref{thm:smooth} has been corrected, the hypotheses of Theorem~\ref{thm:smooth} have been weakened, and Lemma~\ref{le:product} has been moved to Section~\ref{sec:smooth}. The original manuscript can be found on my webpage {\bf http://www.math.ethz.ch/~salamon/publications.html}. } http://www.maia.ub.es/mpej/Vol/10/3.ps http://www.maia.ub.es/mpej/Vol/10/3.pdf http://www.ma.utexas.edu/mpej/Vol/10/3.ps http://www.ma.utexas.edu/mpej/Vol/10/3.pdf http://mpej.unige.ch/mpej/Vol/10/3.ps http://mpej.unige.ch/mpej/Vol/10/3.pdf