We axiomatically characterize two classes of poverty measures which are sensitive to inequality of opportunity – one a strict subset of the other. The proposed indices are sensitive not only to income shortfalls from the poverty line, but also to differences in opportunities faced by people with different pre-determined characteristics, such as race or family background. Dominance conditions are established for each class of measures, and a sub-family of scalar indices, based on a rank-dependent aggregation of type-specific poverty levels, is also introduced. Using household survey data from eighteen European countries in 2005, we find substantial differences in country rankings based on standard FGT indices and on the new opportunity-sensitive indices. Cross-country differences in opportunity-sensitive poverty are decomposed into a level effect; a distribution effect; and a population composition effect.