The dominant conception of brain death as the death of the whole brain constitutes an unstable compromise between the view that a person ceases to exist when she irreversibly loses the capacity for consciousness and the view that a human organism dies only when it ceases to function in an integrated way. I argue that no single criterion of death captures the importance we attribute both to the loss of the capacity for consciousness and to the loss of functioning of the organism as a whole. This is because the person or self is one thing and the human organism is another. We require a separate account of death for each. Only if we systematically distinguish between persons and human organisms will we be able to provide plausible accounts both of the conditions of our ceasing to exist and of when it is that we begin to exist. This paper, in short, argues for a form of mind-body dualism and draws out some of its implications for various practical moral problems.