This book is concerned primarily with the theory of Banach lattices and with linear operators defined on, or with values in, Banach lattices. More general classes of Riesz spaces are considered so long as this does not lead to more complicated constructions or proofs. The intentions for writing this book were twofold. First, there appeared in the literature many results completing the theory extensively. On the other hand, new techniques systematically applied here for the first time lead to surprisingly simple and short proofs of many results originally known as deep. These new methods are purely elementary: they directly yield the Banach lattice versions of theorems which then include the classical theorems in a trivial manner. In particular the book covers: Riesz spaces, normed Riesz spaces, C(K)-and Mspaces, Banach function spaces, Lpspaces, tensor products of Banach lattices, Grothendieck spaces; positive and regular operators, extensions of positive operators, disjointness-preserving operators, operators on L- and M-spaces, kernel operators, weakly compact operators and generalizations, Dunford-Pettis operators and spaces, irreducible operators; order continuity of norms, p-subadditive norms; spectral theory, order spectrum; embeddings of C; the Radon-Nikodym property; measures of non-compactness. This textbook on functional analysis, operator theory and measure theory is intended for advanced students and researchers.