  We review some recent proposals for relativistic models of dark matter in the
context of bimetric gravity. The aim is to solve the problems of cold dark
matter (CDM) at galactic scales, and to reproduce the phenomenology of the
modified Newtonian dynamics (MOND), while still being in agreement with the
standard cosmological model $\Lambda$-CDM at large scales. In this context a
promising alternative is dipolar dark matter (DDM) in which two different
species of dark matter particles are separately coupled to the two metrics of
bigravity and are linked together by an internal vector field. The
phenomenology of MOND then results from a mechanism of gravitational
polarization. Probably the best formulation of the model is within the
framework of recently developed massive bigravity theories. Then the
gravitational sector of the model is safe by construction, but a ghostly degree
of freedom in the decoupling limit is still present in the dark matter sector.
Future work should analyse the cosmological solutions of the model and check
the post-Newtonian parameters in the solar system.
