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