The reduction of poverty and social exclusion is one of the targets of the European Union's 2020 strategy. The appropriateness and success of such a policy require the choice of relevant indicators that not only highlight poverty gaps between countries but also identify the groups of individuals in each country that need particular attention from social policies. The target retained in the European strategy combines three criteria: people living in households below the monetary poverty threshold, poor people “in terms of standard of living” who live in a situation of severe material deprivation, and those who live in households with very low or zero work intensity. We first show that neither the combination nor the intersection of these three criteria produces an adequate measure of the fight against poverty, or an objective for it. We therefore propose an alternative concept, that of “consistent poverty”, which targets people who simultaneously live below the monetary poverty threshold and above a certain level of material deprivation. The special material deprivation module of the EU-SILC 2009 database allows us to examine two versions of this notion of deprivation: the measurement of “severe” deprivation currently used by the European Union, which adopts a threshold with four items, and an alternative measure of “elementary” material deprivation with a three-item threshold. The intersection between our three-item elementary deprivation criterion and the monetary poverty criterion produces more satisfactory results than those obtained by the European Union approach, in terms of both coherency and profile of the population identified.