The compression-expansion theorems of Krasnoselskii represent a main tool in studying nonlinear problems for integral, ordinary differential and partial differential equations and systems. They are used to prove not only the existence of solutions, but also to localize solutions in a conical annulus or other domains of this type.From a theoretical point of view, two major approaches of the subject are known: the first one is in the framework of the fixed point theory and uses essentially Schauder's fixed point theorem and its generalizations, while the second approach uses topological degree theory.The original contributions of Krasnoselskii were given using the first approach and the present book follows the same way.Therefore all our theoretical results are based on fixed point theory.Compared to the literature, our results complement, extend and generalize in several directions the results obtained by R. Legget, L. Williams, R.Avery and others.A consistent part of the book is devoted to applications of the compression-expansion type theorems to several classes of problems: functional- differential equations, systems of ordinary differential equations and others