By definition, an indefinite inner product space is a real or complex vector space together with symmetric (in the complex case: hermitian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values.The main body of the text offers an introducion, with full proofs, to the subject. Prerequisites do not extend much beyond standart facts of linear algebra, topology, Hilbert and Banach space theory.The notes at the end of each chapter contain, besides histirical and other comment relating to the main text, a survey of the literature on more avdanced topics such as the spectral function, groups and algebras of operators, application to differential equations and to quasi-definite functions, etc.