Abstract: We survey various results concerning operator factorization problems. More precisely, we consider the following setting. Let H be a complex Hilbert space, and let B(H) be the algebra of all bounded linear operators on H. For a given subset C of B(H), we are interested in the characterization of operators in B(H) which are expressible as a product of finitely many operators in C and, for each such operator, the minimal number of factors in a factorization. The classes of operators we consider include normal operators, involutions, partial isometries together with their various subclasses, and other miscellaneous classes of operators. Most of the known results are for operators on finite-dimensional spaces or finite matrices. The paper concludes with some applications, due to Hochwald, concerning the uniqueness of the adjoint operation on operators.