  We reinterpret the results of the direct searches for dark matter in terms of
milli-interacting dark particles. The model reproduces the positive results
from DAMA/LIBRA and CoGeNT and is consistent with the absence of signal in the
XENON100, CDMS-II/Ge and LUX detectors. Dark atoms, interacting with standard
atoms through a kinetic mixing between photons and dark photons and a mass
mixing of {\sigma} mesons with dark scalars, diffuse elastically in terrestrial
matter where they deposit all their energy. Reaching underground detectors
through gravity at thermal energies, they form bound states with nuclei of the
active medium by radiative capture, which causes the emission of photons that
produce the observed signals. The parameter space of the model is explored and
regions reproducing the results at the 2{\sigma} level are obtained for each
experiment.
