A NATO Advanced Study Institute on “Non-Linear Evolution and Chaotic Phenomena” was held in Noto, Sicily, June 8–19, 1987. The director was Giovanni Gallavotti (Roma) with co-directors Marcello Anile (Catania) and your author. Other members of the scientific organizing committee included Mitchell Feigenbaum (Rockefeller University) and David Ruelle (IHES). The attendance consisted of 23 invited speakers and about 80 “students”. In this report, I have summarized the talks of the invited speakers who gave between one and three lectures each. In addition, there were contributed seminars, presented during the two-hour afternoon “tutorials” organized by Gallavotti to assist the students' assimilation of the material presented by the principal lecturers. (These students, incidentally, ranged from graduate students through recent Ph.D.'s all the way to an endowed chair professor from the University of Virginia and the former chairman of the mathematics department at Rutgers University!) The contributed seminars are listed by title and author at the end of this report, but here I should mention the special efforts of Jack Dorning (the previously mentioned participant from U.Va.) who contributed four hours' worth of talks on non-linear phenomena and chaos at an introductory level. (He spoke about fixed points, bifurcations, attractors, Poincare maps, return maps, self-similarity, etc.) These lectures, to a large extent, helped overcome the difficulties inherent in a school with students of such widely differing backgrounds and in no little way contributed to a successful learning experience for all concerned.