  We review the properties of neutron matter in the low-density regime. In
particular, we revise its ground state energy and the superfluid neutron
pairing gap, and analyze their evolution from the weak to the strong coupling
regime. The calculations of the energy and the pairing gap are performed,
respectively, within the Brueckner--Hartree--Fock approach of nuclear matter
and the BCS theory using the chiral nucleon-nucleon interaction of Entem and
Machleidt at N$^3$LO and the Argonne V18 phenomenological potential. Results
for the energy are also shown for a simple Gaussian potential with a strength
and range adjusted to reproduce the $^1S_0$ neutron-neutron scattering length
and effective range. Our results are compared with those of quantum Monte Carlo
calculations for neutron matter and cold atoms. The Tan contact parameter in
neutron matter is also calculated finding a reasonable agreement with
experimental data with ultra-cold atoms only at very low densities. We find
that low-density neutron matter exhibits a behavior close to that of a Fermi
gas at the unitary limit, although, this limit is actually never reached. We
also review the properties (energy, effective mass and quasiparticle residue)
of a spin-down neutron impurity immersed in a low-density free Fermi gas of
spin-up neutrons already studied by the author in a recent work where it was
shown that these properties are very close to those of an attractive Fermi
polaron in the unitary limit.
