Two closely related issues are addressed in this thesis. The first of these issues is whether Lockean criteria can withstand criticisms based on cases such as Wiggins' brain bisection case. The problem here is that the memory relation, or in general any relation of psychological continuity, seems to be a many-one relation and hence not suitable as a criterion of identity for persons. The second issue involves the question of what we ought to say about such cases. The problem here (...) is that, intuitively, psychological continuity seems to be a very good criterion of personal identity. When one person is psychologically continuous with a future person, we want to say that the one person has survived--as that future person. ;Several contemporary philosophers have addressed themselves to these issues. In Chapter I of the thesis, I consider the theory of identity suggested by Baruch Brody. He rejects Lockean criteria, and instead uses his own perfectly general theory as a criterion of identity. I argue that the theory he suggests is of limited value, and that the arguments he provides against Lockean criteria are question-begging. In Chapter II, I develop a specific Lockean criterion; in Chapter III, I consider and finally reject particular arguments against this criterion which are based on the brain bisection cases. In this chapter, I consider work by David Wiggins, Bernard Williams, and Roland Puccetti. In Chapter IV, I turn to the defense of Lockean criteria suggested by John Perry and David Lewis. Finally, in Chapter V, I develop my own views on the brain bisection case, and concluding that Lockean criteria of a certain sort remain defensible. (shrink)