Poverty alleviation still occupies a top position in many countries’ economic and political agendas. Although income has a central role in defining poverty and well-being, poverty should, more generally, be framed as a multidimensional phenomenon. Examples of these dimensions are housing, literacy and life expectancy. The scope of this chapter is to present a discussion on the axiomatic approach to the measurement of multidimensional poverty and material deprivation, where the former should encompass all dimensions of well-being, while the latter concerns only the functioning failures related to material living conditions. The chapter ends with a discussion of the main indices proposed in the literature on the measurement of these phenomena.