This book is on Kolmogorov-Arnol'd-Moser theory for quasi-periodic tori in dynamical systems. It gives an up-to-date report on the role parameters play for persistence of such tori, typically occuring on Cantor sets of positive Hausdorff measure inside phase and parameter space. The cases with preservation of symplectic or volume forms or time-reversal symmetries are included. The concepts of Whitney-smoothness and Diophantine approximation of Cantor sets on submanifolds of Euclidean space are treated, as well as Bruno's theory on analytic continuation of tori. Partly this material is new to Western mathematicians. The reader should be familiar with dynamical systems theory, differential equations and some analysis. The book is directed to researchers, but its entrance level is introductory.

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