Low-energy absorption cross sections for various particles falling into extreme non-dilatonic branes are calculated using string theory and world-volume field theory methods. The results are compared with classical absorption by the corresponding gravitational backgrounds. For the self-dual three-brane, earlier work by one of us demonstrated precise agreement of the absorption cross sections for the dilaton, and here we extend the result to Ramond-Ramond scalars and to gravitons polarized parallel to the brane. In string theory, the only absorption channel available to dilatons and Ramond-Ramond scalars at leading order is conversion into a pair of gauge bosons on the three-brane. For gravitons polarized parallel to the brane, scalars, fermions and gauge bosons all make leading-order contributions to the cross section, which remarkably add up to the value predicted by classical gravity. For the two-brane and five-brane of M-theory, numerical coefficients fail to agree, signaling our lack of a precise understanding of the world-volume theory for large numbers of coincident branes. In many cases, we note a remarkable isotropy in the final state particle flux within the brane. We also consider the generalization to higher partial waves of minimally coupled scalars. We demonstrate agreement for the three-brane at l = 1 and indicate that further work is necessary to understand l > 1.