The aim of this thesis is to introduce and elaborate a new conception of the relation between wholes and parts. Wholes, I propose, can be conceived of as 'Unities', in contrast to their currently familiar conception as 'sums'. My starting point is the distinction which Plato draws in Theaetetus (203c-205a) between two conceptions of a complex entity, and the contrast between Unities and sums emerges along lines suggested by Plato's distinction. Part I is preoccupied with general conceptual and historical background. Some fundamental assumptions which might appear to constrain any theory of wholes and parts are placed in perspective, thus preparing the grounds for their rejection in the context of a theory of Unities. In Part II the conception of wholes as sums is presented, and it is extensively argued that modern discussions of wholes generally presuppose this conception. This presupposition is shared not only by authors who subscribe to the 'classical' mereological theories of Lesniewski and Goodman, but also by theorists of holistic sympathies (i.e. theorists who make use, for example, of the notion of an organic whole, or of a Gestalt) who rely on 'neoclassical' theories. It is urged that this conception suffers from serious, fundamental difficulties and drawbacks. In Part III the conception of wholes as Unities is introduced. A theory of Unities is laid down in systematic, formal detail, and the points of divergence from presuppositions of traditional theories are discussed. It is shown how in conceiving of concrete entities (of certain types) as Unities one is free from many of difficulties which beset their conception as sums. Finally, it is shown how the theory of Unities provides a powerful tool for resolving some central metaphysical puzzles concerning concrete entities, puzzles associated with preservation of identity in the face of loss or gain of parts. In addition, it is pointed out that there is scope for a wider application of the theory of Unities - e.g. in the theory of events.