In this thesis we study operator ideals on ordered Banach spaces such as Banach lattices, $C^*$-algebras, and noncommutative function spaces. The first part of this work is concerned with the domination problem: the relationship between order and algebraic ideals of operators. Fremlin, Dodds and Wickstead described all Banach lattices on which every operator dominated by a compact operator is always compact. First, we show that even if the dominated operator is not compact it still belongs to a relatively small class of operators, namely, the ideal of inessential operators. A similar question is studied for strictly singular operators. In particular, we show that the cube of every operator, dominated by a strictly singular operator, is inessential. Then we provide a complete solution of the domination problem for compact and weakly compact operators acting between $C^{*}$-algebras and noncommutative function spaces. Finally, we consider the domination problem for weakly compact operators acting on general noncommutative function spaces. The second part is devoted to the operator ideal structure of the algebra of all linear bounded operators on a Banach space. First, we investigate the existence of non-trivial proper ideals on Lorentz sequence spaces and characterize some of them. Second, we look at the coincidence of some classical operator ideals, such as of compact, strictly singular, innesential, and Dunford-Pettis operators acting on noncommutative $L_p$-spaces. In particular, we obtain a characterization of strictly singular and inessential operators acting either between discrete noncommutative $L_p$-spaces or $L_p$-spaces, associated with a hyperfinite von Neumann algebras with finite trace.