This thesis is an investigation into several proposed ways of increasing the supply of organs for transplantation purposes. It starts from a consideration of two sorts of donors which are already in use: humans declared dead according to brain-based criteria and nonhuman animals. In the first chapter I attempt to provide a theoretical underpinning for the brainstem criterion of brain death. I also criticise certain rival theories. In the second chapter I consider what constraints govern what we may do to the newly dead body - the fact that someone is dead does not mean we may treat their corpse as we please. I consider Professor John Harris' strong `opting out' policy, and argue against it. I briefly discuss other alternatives to the UK's present `opting in' system - `presumed consent' and `required request'. And I consider the problems generated by the proposed use as donors of another category of human cadaver - people declared dead according to cardiorespiratory criteria. The third chapter is an investigation into xenotransplantation - the use of organs from nonhuman animals. I argue that using animals `just because' they are animals is ethically indefensible. Instead I attempt to justify the killing of (some) animals as organ donors on the grounds that their interest in continued life is weaker than that of a human. But this has some counterintuitive implications concerning arational members of our species. In the final chapter I discuss the possible use of the `worst off' category of arational humans - the permanently and irreversibly unconscious. I argue that we have moral obligations to these people which do not stem only from a consideration of their interest in life. However, I think that we may use such people as donors under specialised conditions: if we are as certain as we can be that they are irreversibly unconscious, if their family (and ideally the donor too) have requested it, and so long as it can be done without distress to the public. Under no other circumstances do I think humans ought to be killed for organs.