The proportion of poor people living in neighborhoods where poverty is highly concentrated is widely accepted as a policy-relevant measure of urban poverty in large metro areas. We challenge this view by developing new measures of urban poverty that i) capture aspects of the incidence and distribution of poverty across neighborhoods and ii) are consistent with the idea that poor people living in high-poverty neighborhoods face a double welfare burden of poverty. We demonstrate that there is only a measure that is consistent with a parsimonious axiomatic model for urban poverty. Panel variations of this measure are additively decomposed into the contribution of demographic, spatial and neighborhood-level poverty convergence effects. We collect new evidence of heterogenous patterns and trends of urban poverty across American metro areas over the last 35 years. Reduced form models allow to recover the implications of (income) sorting, affordable housing and of rising gentrification on different components of urban poverty across American cities.