In the past several decades many significant results in averaging for systems of ODE's have been obtained. These results have not attracted a tention in proportion to their importance, partly because they have been overshadowed by KAM theory, and partly because they remain widely scattered - and often untranslated - throughout the Russian literature. The present book seeks to remedy that situation by providing a summary, including proofs, of averaging and related techniques for single and multiphase systems of ODE's. The first part of the book surveys most of what is known in the general case...