The book addresses the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fundamental localization principle underlying these solutions has mostly gone unnoticed. This ignorance of the general principle has often required the use of various artificial tricks and hindered the resolution of new important problems in index theory. So far, the localization principle has been scarcely covered in journal papers and not at all in monographs. This book aims to fill that gap. It is the first and only monograph on the topic. Both the general localization principle and its applications to specific existing and new problems are discussed. The book will be of interest to working mathematicians as well as graduate and postgraduate university students specializing in differential equations and related topics.