  We explore some of the consequences of Dark Matter-photon interactions on
structure formation, focusing on the evolution of cosmological perturbations
and performing both an analytical and a numerical study. We compute the cosmic
microwave background anisotropies and matter power spectrum in this class of
models. We find, as the main result, that when Dark Matter and photons are
coupled, Dark Matter perturbations can experience a new damping regime in
addition to the usual collisional Silk damping effect. Such Dark Matter
particles (having quite large photon interactions) behave like Cold Dark Matter
or Warm Dark Matter as far as the cosmic microwave background anisotropies or
matter power spectrum are concerned, respectively. These Dark Matter-photon
interactions leave specific imprints at sufficiently small scales on both of
these two spectra, which may allow to put new constraints on the acceptable
photon-Dark Matter interactions. Under the conservative assumption that the
abundance of 10^12 M_sol galaxies is correctly given by Cold Dark Matter, and
without any knowledge of the abundance of smaller objects, we obtain the limit
on the ratio of the Dark Matter-photon cross section to the Dark Matter mass
sigma_{gamma-DM} / m_DM < 10^-6 sigma_Thomson / 100 GeV \sim 6 * 10^-33 cm^2
GeV^-1 .
