Coaugmented homotopy idempotent localization functors -- Augmented homotopy idempotent functors -- Commutation rules for ?, Lf and CWA, preservation of fibrations and cofibrations -- Dold-Thom symmetric products and other colimits -- General theory of fibrations, GEM error terms -- Homological localization nearly preserves fibrations -- Classification of nullity and cellular types of finite p-torsion suspension spaces -- v 1-periodic spaces and K-theory -- Cellular inequalities.

In this monograph we give an exposition of some recent development in homotopy theory. It relates to advances in periodicity in homotopy localization and in cellular spaces. The notion of homotopy localization is treated quite generally and encompasses all the known idempotent homotopy functors. It is applied to K-theory localizations, to Morava-theories, to Hopkins-Smith theory of types. The method of homotopy colimits is used heavily. It is written with an advanced graduate student in topology and research homotopy theorist in mind.