This paper offers an axiomatic characterization of two classes of poverty measures that 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 the opportunities faced by people with different predetermined 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. In empirical analysis using household survey data from eighteen European countries in 2005, substantial differences in country rankings based on standard Foster-Greer-Thorbecke indices and on the new opportunity-sensitive indices are found. Cross-country differences in opportunity-sensitive poverty are decomposed into a level effect, a distribution effect, and a population composition effect.