We show that, contrary to the situation in the vacuum, a Majorana neutrino can have electric and magnetic dipole moments in a medium. This is because of new contributions, equal for a particle and its antiparticle, that can arise only in a material background. For Dirac neutrinos, these contributions make the magnitudes of the dipole moments of the particle and the antiparticle unequal. We discuss the conditions which give rise to such effects, with particular attention to the role played by the discrete symmetries {ital C}, {ital P}, and {ital T}.