The notions of a locally α-presentable and locally α-generated categories are introduced, where α is a regular cardinal. The properties of these categories are studied extensively, in particular their close relationship with other types of categories. Also the subclasses of topos, algebraic categories and locally α-noetherian categories are investigated in detail. A “classification” of locally α-presentable, locally α-generated categories, locally α-noetherian categories and algebraic categories is given.