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The study of the spontaneous formation of nanostructures in single crystals of several compounds is now a major area of research in strongly correlated electrons. These structures appear to originate in the competition of phases. The book addresses nanoscale phase separation, focusing on the manganese oxides known as manganites that show the colossal magnetoresistance (CMR) effect which has potential relevance for device applications. It is argued that the nanostructures are at the heart of the CMR phenomenon. The book contains updated information on manganite research directed to experts, both theorists and experimentalists. However, graduate students or postdocs will find considerable introductory material, including elements of computational physics. The book also briefly addresses compounds where similar nanostructures have been unveiled, such as high-temperature superconductors and Eu-based semiconductors. Contributions by several distinguished researchers are also included.