This paper presents a study of the equation of state (EOS) of nucleon matter and the structure of neutron stars using the Argonne v18 (A18) two-nucleon interaction and the Urbana IX (UIX) three-nucleon interaction. The A18 interaction provides an excellent fit to nucleon-nucleon scattering data, leading to a neutron star mass limit of 1.67 solar masses. Including relativistic boost corrections and three-nucleon interactions increases this limit to 1.80 and 2.20 solar masses. The inclusion of three-nucleon interactions predicts a phase transition in neutron star matter to a phase with neutral pion condensation at a baryon number density of ~0.2 fm⁻³. Neutron stars predicted by these models have a thin layer with a thickness of tens of meters, where the density changes rapidly from normal to condensed phases. The material in this layer is a mixture of the two phases. The paper also investigates the possibility of dense nucleon matter having an admixture of quark matter, described using the bag model equation of state. Neutron stars of 1.4 solar masses do not appear to have quark matter admixtures in their cores, but the heaviest stars are predicted to have cores consisting of a mixture of quark and nucleon matter. These admixtures reduce the maximum mass of neutron stars from 2.20 to 2.02 (1.91) solar masses for bag constants B = 200 (122) MeV/fm³. Stars with pure quark matter in their cores are found to be unstable. The paper also considers the possibility that matter is maximally incompressible above a certain density, and shows that realistic models of nuclear forces limit the maximum mass of neutron stars to be below 2.5 solar masses. The effects of phase transitions on the composition of neutron star matter and its adiabatic index Γ are discussed. The study uses variational chain summation (VCS) methods and includes relativistic boost corrections to the A18 interaction. The results show that the inclusion of three-nucleon interactions and relativistic corrections significantly affects the structure and properties of neutron stars. The paper concludes that the A18+UIX model provides a more accurate description of neutron star structure and properties compared to previous models.This paper presents a study of the equation of state (EOS) of nucleon matter and the structure of neutron stars using the Argonne v18 (A18) two-nucleon interaction and the Urbana IX (UIX) three-nucleon interaction. The A18 interaction provides an excellent fit to nucleon-nucleon scattering data, leading to a neutron star mass limit of 1.67 solar masses. Including relativistic boost corrections and three-nucleon interactions increases this limit to 1.80 and 2.20 solar masses. The inclusion of three-nucleon interactions predicts a phase transition in neutron star matter to a phase with neutral pion condensation at a baryon number density of ~0.2 fm⁻³. Neutron stars predicted by these models have a thin layer with a thickness of tens of meters, where the density changes rapidly from normal to condensed phases. The material in this layer is a mixture of the two phases. The paper also investigates the possibility of dense nucleon matter having an admixture of quark matter, described using the bag model equation of state. Neutron stars of 1.4 solar masses do not appear to have quark matter admixtures in their cores, but the heaviest stars are predicted to have cores consisting of a mixture of quark and nucleon matter. These admixtures reduce the maximum mass of neutron stars from 2.20 to 2.02 (1.91) solar masses for bag constants B = 200 (122) MeV/fm³. Stars with pure quark matter in their cores are found to be unstable. The paper also considers the possibility that matter is maximally incompressible above a certain density, and shows that realistic models of nuclear forces limit the maximum mass of neutron stars to be below 2.5 solar masses. The effects of phase transitions on the composition of neutron star matter and its adiabatic index Γ are discussed. The study uses variational chain summation (VCS) methods and includes relativistic boost corrections to the A18 interaction. The results show that the inclusion of three-nucleon interactions and relativistic corrections significantly affects the structure and properties of neutron stars. The paper concludes that the A18+UIX model provides a more accurate description of neutron star structure and properties compared to previous models.