  We systematically study light (< few GeV) Dark Matter (DM) models that
thermalize with visible matter through the Higgs portal and identify the
remaining gaps in the viable parameter space. Such models require a comparably
light scalar mediator that mixes with the Higgs to avoid DM overproduction and
can be classified according to whether this mediator decays (in)visibly. In a
representative benchmark model with Dirac fermion DM, we find that, even with
conservative assumptions about the DM-mediator coupling and mass ratio, the
regime in which the mediator is heavier than the DM is fully ruled out by a
combination of collider, rare meson decay, and direct detection limits; future
and planned experiments including NA62 can further improve sensitivity to
scenarios in which the Higgs portal interaction does not determine the DM
abundance. The opposite, regime in which the mediator is lighter than the DM
and the latter annihilates to pairs of visibly-decaying mediators is still
viable, but much of the parameter space is covered by rare meson decay,
supernova cooling, beam dump, and direct detection constraints. Nearly all of
these conclusions apply broadly to the simplest variations (e.g. scalar or
asymmetric DM). Future experiments including SHiP, NEWS, and Super-CDMS SNOLAB
can greatly improve coverage to this class of models.
