There has been a growing interest in what have come to be termed "multidimensional indices of poverty." Advocates for these new indices correctly point out that command over market goods is not all that matters to peoples' well-being, and that other factors need to be considered when quantifying the extent of poverty and informing policy making for fighting poverty. However, the author argues that there are two poorly understood issues in assessing these indices. First, does one believe that any single index can ever be a sufficient statistic for poverty assessments? Second, when aggregation is called for, should it be done in the space of "attainments," using prices when appropriate, or that of "deprivations," using weights set by the analyst? The paper argues that the goal for future poverty monitoring efforts should be to develop a credible set of multiple indices, spanning the dimensions of poverty most relevant to a specific setting, rather than a single multidimensional index. When weights are needed, they shouldn't be set solely by an analyst measuring poverty. Rather, they should be, as much as possible, consistent with well-informed choices made by poor people.