We develop in this paper a systematic study of the stochastic dominance ordering in spaces of measures. We collect and present in an orderly fashion, results that are spread out in the Applied Probability and Mathematical Economics literature, and extend most of them to a somewhat broader framework. Several original contributions are made on the way. We provide a sharp characterization of conditions that permit an equivalent definition of stochastic dominance by means of continuous and monotone functions. When the preorder of the original space is closed, we offer an extremely simple equivalent characterization of stochastic dominance, a result of which we have found no parallel in literature. We develop original methods that shed light into the inheritance of the antisymmetric property by the stochastic dominance ordering. We study how the topological properties of the preorder translate to the stochastic dominance preorder. A class of spaces in which monotone and continuous functions are convergence-determining is described. Finally, conditions are given that guarantee that (order) bounded stochastically monotone nets, have a limit point.We develop in this paper a systematic study of the stochastic dominance ordering in spaces of measures. We collect and present in an orderly fashion, results that are spread out in the Applied Probability and Mathematical Economics literature, and extend most of them to a somewhat broader framework. Several original contributions are made on the way. We provide a sharp characterization of conditions that permit an equivalent definition of stochastic dominance by means of continuous and monotone functions. When the preorder of the original space is closed, we offer an extremely simple equivalent characterization of stochastic dominance, a result of which we have found no parallel in literature. We develop original methods that shed light into the inheritance of the antisymmetric property by the stochastic dominance ordering. We study how the topological properties of the preorder translate to the stochastic dominance preorder. A class of spaces in which monotone and continuous functions are convergence-determining is described. Finally, conditions are given that guarantee that (order) bounded stochastically monotone nets, have a limit point.