The state space factorization method is systematically used and developed further for various classes of matrix and operator functions Emphasis is put on canonical factorization problems, including spectral and J-spectral factorizations problems and related Ricatti equations Elements of H-infinity control theory and the related Nehari approximation problem are covered Applications concern elements of H-infinity control theory and the related Nehari approximation problem, problems in mathematical analysis (inversion problems for singular integral equations and Wiener-Hopf integral equations, related Riemann Hilbert problems), and problems from mathematical physics (linear transport theory) A large part the book deals with rational matrix functions only Includes supplementary material: sn.pub/extras