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 Λ-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 analyze the cosmological solutions of the model and check the post-Newtonian parameters in the solar system.