The book treats Boolean valued analysis. This term signifies the technique of studying properties of an arbitrary mathematical object by means of comparison between its representations in two different set-theoretic models whose construction utilizes principally distinct Boolean algebras. As these models, we usually take the classical Cantorian paradise in the shape of the von Neumann universe and a specially-trimmed Boolean valued universe in which the conventional set- theoretic concepts and propositions acquire bizarre interpretations. Exposition focuses on the fundamental properties of order bounded operators in vector lattices. This volume is intended for the classical analyst seeking new powerful tools and for the model theorist in search of challenging applications of nonstandard models of set theory.