Abstract:  In classical matrix theory, a matrix can be written in upper triangular form, with help of its invariant subspaces. An important goal is to find such upper triangular forms and invariant subspaces (or prove they don't exist!) for linear operators on infinite dimensional spaces. We will review some older results on these topics and then describe some remarkable invariant subspaces found by Haagerup and Schultz for operators in finite von Neumann algebras, and how they can be used to find upper triangular forms.