Book section: This paper generalizes the poverty ordering criteria available for single dimensional income poverty to the case of multidimensional welfare attributes. A set of properties to be satisfied by multidimensional poverty measures is first discussed. Then general classes of poverty measures based on these properties are defined. Finally, dominance criteria are derived such that a distribution of multidimensional attributes exhibits less poverty than another for all multidimensional poverty indices belonging to a given class. These criteria may be seen as a generalization of the single dimensional poverty-line criterion. However, it turns out that the way this generalization is made depends on whether attributes are complements or substitutes.