Modal set theory draws in particular upon contemporary modal logic, the logic of necessity and possibility. One simple and obvious motivation for modal set theory is the fact that, from a realist perspective that takes the existence of sets seriously, sets have philosophically interesting modal properties. This chapter focuses on the technical development of modal set theory and its intuitive motivations without any close critical attention paid to surrounding philosophical questions. It devotes to the development of modal set theory with regard to a rather more directed inquiry into both the nature and structure of sets that is motivated in particular by the attractive prospect of a satisfying explanation of Russells Paradox. The years 18971903 saw the emergence of a string of related paradoxes concerning the notions of number, set, class, property, proposition, and truth. Metaphysical modality would be unsuitable for our present purposes because pure sets are taken to exist of metaphysical necessity if at all.