  If dark matter (DM) has non-zero direct or transition, electric or magnetic
dipole moment then it can scatter nucleons electromagnetically in direct
detection experiments. Using the results from experiments like XENON, CDMS,
DAMA and COGENT we put bounds on the electric and magnetic dipole moments of
DM. If DM consists of Dirac fermions with direct dipole moments, then DM of
mass less than 10 GeV is consistent with the DAMA signal and with null results
of other experiments. If on the other hand DM consists of Majorana fermions
then they can have only non-zero transition moments between different mass
eigenstates. We find that Majorana fermions with mass m_\chi > 38 GeV and mass
splitting of the order of (50-200) keV can explain the DAMA signal and the null
observations from other experiments and in addition give the observed relic
density of DM by dipole-mediated annihilation. This parameter space for the
mass and for dipole moments is allowed by limits from L3 but may have
observable signals at LHC.
