  Dark matter self-interactions are a well-motivated solution to the
core-vs.-cusp and the too-big-to-fail problems. They are commonly induced by
means of a light mediator, that is also responsible for the dark matter
freeze-out in the early universe. Motivated by the fact that such scenario is
excluded in its simplest realizations, we will discuss the possibility that the
relic density of a self-interacting dark matter candidate can proceed from the
freeze-out of only annihilations into SM particles. We will argue that scalar
and Majorana dark matter in the mass range of 10 to 500 MeV, coupled to a
slightly heavier massive gauge boson, are the only candidates in agreement with
multiple current experimental constraints. We will also discuss prospects of
establishing or excluding these two scenarios in future experiments.
