  We examine cosmological, astrophysical and collider constraints on thermal
dark matter (DM) with mass mX in the range 1 MeV to 10 GeV. Cosmic microwave
background (CMB) observations, which severely constrain light symmetric DM, can
be evaded if the DM relic density is sufficiently asymmetric. CMB constraints
require the present anti-DM to DM ratio to be less than 2*10^{-6} (10^{-1}) for
DM mass mX = 1 MeV (10 GeV) with ionizing efficiency factor f ~ 1. We determine
the minimum annihilation cross section for achieving these asymmetries subject
to the relic density constraint; these cross sections are larger than the usual
thermal annihilation cross section. On account of collider constraints, such
annihilation cross sections can only be obtained by invoking light mediators.
These light mediators can give rise to significant DM self-interactions, and we
derive a lower bound on the mediator mass from elliptical DM halo shape
constraints. We find that halo shapes require a mediator with mass mphi > 4 *
10^{-2} MeV (40 MeV) for mX = 1 MeV (10 GeV). We map all of these constraints
to the parameter space of DM-electron and DM-nucleon scattering cross sections
for direct detection. For DM-electron scattering, a significant fraction of the
parameter space is already ruled out by beam-dump and supernova cooling
constraints.
