For the last thirty years or so, there has been a search underway for a theory that canaccommodate our intuitions in regard to moral duties to future generations. The object ofthis search has proved surprisingly elusive. The classical moral theories in the literature allhave perplexing implications in this area. Classical Utilitarianism, for instance, implies thatit could be better to expand a population even if everyone in the resulting populationwould be much worse off than in the original. The main problem has been to find an adequate population theory, that is, a theoryabout the moral value of states of affairs where the number of people, the quality of theirlives, and their identities may vary. Since, arguably, any reasonable moral theory has totake these aspects of possible states of affairs into account when determining the normativestatus of actions, the study of population theory is of general import for moral theory. A number of theories have been proposed in the literature that purport to avoidcounter-intuitive implications such as the one mentioned above. The suggestions arediverse: introducing novel ways of aggregating welfare into a measure of value, revising thenotion of a, life worth living, questioning the way we can compare and measure welfare,counting people's welfare differently depending on the temporal location or the modalfeatures of their lives, and challenging the logic of axiological and normative concepts. Weinvestigate the concepts and assumptions involved in these theories as well as theirimplications for population theory. In our discussion, we propose a number of intuitively appealing and logically weakadequacy conditions for an acceptable population theory. Finally, we consider whether it ispossible to find a theory that satisfies all of these conditions. We prove that no such theory exists.