Let X =(X, ?) be a topological space and let X be the union of two (?, respectively) disjoint dense subsets, then X is called resolvable (?-resolvable, respectively). A space which is not resolvable (not ?-resolvable, respectively) is called irresolvable (?-irresolvable, respectively). The aim of present thesis is to determine conditions under which a space is a resolvable space. This thesis is divided into five chapters. In the first chapter we mention required definitions and theorems. The methods needed to study irresolvable spaces are those needed to study expansion of spaces. For this reason, the second chapter is devoted to the development of the theory of such expansions. We investigate properties of spaces which are preserved under arbitrary expansions, such as the property of being a ace, ace, Hausdorff space and Urysohn space. In considering various types of expansions, we prove the existence of a particular expansion, called a maximal expansion, enjoying a number of certain special properties. Maximal space is essential in the construction of irresolvable spaces. Also, we consider various properties of contraction, which is the reverse operation with respect to expansion. The Third chapter contains a study of irresolvable spaces. The existence of totally disconnected Urysohn spaces, irresolvable connected aces and totally disconnected completely regular spaces are proved directly from the expansion theory. Some irresolvable spaces enjoy a very strong disconnectivity property, which we investigate in detail. In the fourth chapter, it is shown that metric spaces, compact Hausdorff spaces, Hausdorff spaces satisfying the first countability axiom, and various other special spaces are all resolvable. We also obtain a reduction showing that only aces, and in fact only compact aces are, need to be considered in dealing with the resolution problem. In the final chapter we study under what conditions a locally connected space and ace can be resolved