We construct explicit models of particle dark matter where the attractive force in the dark matter sector creates a narrow near-threshold resonance that qualitatively changes the energy dependence of the annihilation cross section. In these models, the resonant enhancement of the dark matter annihilation can easily source the excess of energetic leptons observed by experiments on PAMELA and FERMI satellites. The distinct feature of these models is that by construction the enhancement of the annihilation cross section shuts off when the dark matter velocity falls below the typical Milky Way values, thus automatically satisfying constraints on dark matter annihilation imposed by the CMB anisotropies and gamma ray constraints from satellite galaxies.

20 pages, 10 figures