Theoretical models of the equation of state (EOS)
of neutron-star matter (starting with the crust
and ending at the densest region of the stellar core) are reviewed. Apart from
a broad set of baryonic EOSs, strange quark matter, and even more exotic (abnormal
and Q-matter) EOSs are considered. Results of calculations of Mmax
for non-rotating neutron stars and exotic compact stars are reviewed, with
particular emphasis on the dependence on the dense-matter EOS. Rapid rotation
increases Mmax, and this effect is studied for both neutron stars and exotic
stars. Theoretical results are then confronted with measurements of masses of neutron
stars in binaries, and the consequences of such a confrontation and their possible impact
on the theory of dense matter are discussed.