  In this review, we summarise the main features of the BPS Skyrme model which
provides a physically well-motivated idealisation of atomic nuclei and nuclear
matter: 1) it leads to zero binding energies for classical solitons (while
realistic binding energies emerge owing to the semiclassical corrections, the
Coulomb interaction and isospin breaking); 2) it describes a perfect
non-barotropic fluid already at the microscopic (field theoretical) level which
allows to study thermodynamics beyond the mean-field limit. These properties
allow for an approximate but analytical calculation of binding energies of the
most abundant nuclei, for a determination of the equation of state of
skyrmionic matter (both in the full field theory and in a mean-field
approximation) as well as the description of neutron stars as Skyrme solitons
with a very good agreement with available observational data. All these results
suggest that the proper low energy effective model of QCD should be close to
the BPS Skyrme model in a certain sense (a "near-BPS Skyrme model"), with a
prominent role played by the BPS part.
