Methods of synthesis, properties and transformations of fullerene peroxides are considered and systematized for the first time. It is shown that the chemistry of fullerene peroxides is a new approach to functionalization of fullerenes, which has been intensively developing since 2002. Methods of synthesis, mechanisms of formation and reactions of C60 and C70 alkyl peroxides with or without epoxide moieties are discussed. Transformations of fullerene peroxides affording a wide range of fullerene derivatives containing, as addends, halogen or sulfur atoms; epoxide, dioxolane, thiirane, crown ether, aziridine and dioxetane rings, as well as hydroxyl, alkoxyl and carbonyl groups, are considered. Special attention is focused on reactions constituting the basis of a new approach — so-called molecular surgery, which enables the synthesis of open-cage fullerene derivatives. It has been demonstrated that such compounds are good candidates for designing photovoltaic cells and carriers of drugs and radionuclides (for radiopharmaceuticals). The bibliography includes 130 references.