Triangular algebras and nest algebras are two important classes of non-selfadjoint operator algebras. In this book, the author uses the new depth of understanding which the similarity theory for nests has opened up to study ideals of nest algebras. In particular, a unique largest diagonal-disjoint ideal is identified for each nest algebra. Using a construction proposed by Kadison and Singer, this ideal can be used to construct new maximal triangular algebras. These new algebras are the first concrete descriptions of maximal triangular algebras that are not nest algebras.