Translation of the V.I. Arnold paper From Superpositions to KAM Theory (Vladimir Igorevich Arnold. Selected-60, Moscow: PHASIS, 1997, pp. 727-740)

V. I. Arnold (12 June 1937-3 June 2010) published several papers where he described, in the form of recollections, his two earliest research problems (superpositions of continuous functions and quasi-periodic motions in dynamical systems), the main results and their interrelations: [A1], then [A2] (reprinted as [A4, A6]), and [A3] (translated into English by the author as [A5]). The first exposition [A1] has never been translated into English; however, it contains many details absent in the subsequent articles. It seems therefore that publishing the English translation of the paper [A1] would not be superfluous. What follows is this translation. In many cases, the translator gives complete bibliographic descriptions of various papers mentioned briefly in the original Russian text. The English translations of papers in Russian are also pointed out where possible. A related material is contained also in Arnold's recollections On A.N. Kolmogorov. Slightly different versions of these reminiscences were published several times in Russian and English [A7-A12]. The early history of KAM theory is also discussed in detail in the recent brilliant semi-popular book [A13].