The study of nest algebras is the study of triangular forms for operators and operator algebras. In this chapter, we first study some basic properties of nest algebras. The bulk of the chapter is devoted to a proof of the Similarity theorem, which describes precisely when two nest algebras acting on a separable Hilbert space are similar. This is followed by a number of applications. We recall from Definition  12.4.6 and Remark  12.4.7 that if \({\mathcal {A}}\) is an algebra of operators, then \(\operatorname {Lat}{\mathcal {A}}\) is a complete lattice of its invariant subspaces. Conversely, if \({\mathcal {L}}\) is a collection of subspaces, then \(\operatorname {Alg}{\mathcal {L}}\) is the unital wot-closed operator algebra consisting of all operators leaving each element of \({\mathcal {L}}\) invariant.