We say that admits a standard triangularization if we can find a totally ordered set and an increasing family of (standard) orthogonal projections in which satisfy (a) is invariant under for all (where denotes the range of the operator ), and (b) the family is a maximal increasing family of standard, invariant projections for . In the event that where refers to the weak operator topology in , we say that admits a multiplicity-free standard triangularization with respect to the maximal abelian, self-adjoint algebra (i.e., a masa) . In this thesis, we establish conditions on an operator which guarantee the existence of a standard, multiplicity-free triangularization of .