The present book deals with factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, theory of job scheduling in operations research. The book systematically employs a geometric principle of factorization which has its origins in the state space theory of linear input-output systems and in the theory of characteristic operator functions. This principle allows one to deal with different factorizations from one point of view. Covered are canonical factorization, minimal and non-minimal factorizations, pseudo-canonical factorization, and various types of degree one factorization.Considerable attention is given to the matter of stability of factorization which in terms of the state space method involves stability of invariant subspaces.invariant subspaces.