The focus of this dissertation is the study of Kadison-Singer algebras. We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. We have classified diagonal-trivial maximal non selfadjoint subalgebras of matrix algebras with lower dimensions. We show that Alg(F₃) is maximal in the class of reflexive subalgebras of B( H), where the restriction of the diagonal of each such subalgebra in L_G3 is trivial. We show that the reflexive algebras Alg(L) given by a double triangular lattice L in M (a factor of type II1) is maximal non selfadjoint in the class of all weak operator closed subalgebras with respect to its diagonal if L generates M. Our method can be used to prove similar results in finite dimensional matrix algebras. As a consequence, we give a new proof to Hou's result, that is, each reflexive algebra determined by a double triangular lattice of projections in a matrix algebra has the diagonal maximality if the double triangular lattice of projections generated the whole matrix algebra.