This chapter provides theoretical foundations for the Prioritarianism in Practice volume, by clarifying the features of prioritarian social welfare functions (SWFs). A prioritarian SWF sums up individuals’ well-being numbers plugged into a strictly increasing and strictly increasing transformation function. Prioritarian SWFs, like the utilitarian SWF, fall within the “generalized-utilitarian” class of SWFs.Generalized-utilitarian SWFs are additive and, hence, especially tractable for purposes of policy analysis.The chapter reviews the axiomatic properties of generalized-utilitarian SWFs and, specifically, of prioritarian SWFs. Prioritarianism satisfies the Pigou-Dalton axiom (a pure, gap-diminishing transfer of well-being from a better-off to a worse-off person is an ethical improvement), while utilitarianism does not. Pigou-Dalton is the axiomatic expression of the fact that a prioritarian SWF gives extra weight (priority) to well-being changes affecting worse-off individuals. The chapter also discusses the informational requirements of prioritarian SWFs (as regards interpersonal well-being comparisons).It reviews the various methodologies for applying a prioritarian SWF under uncertainty. And it describes the two main subfamilies of prioritarian SWFs, namely Atkinson and Kolm-Pollak SWFs.