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Properties of dense nucleon matter and the structure of neutron stars are studied using variational chain summation methods and the new Argonne ${v}_{18}$ two-nucleon interaction, which provides an excellent fit to all of the nucleon-nucleon scattering data in the Nijmegen database. The neutron star gravitational mass limit obtained with this interaction is 1.67${M}_{\ensuremath{\bigodot}}.$ Boost corrections to the two-nucleon interaction, which give the leading relativistic effect of order ${(v/c)}^{2},$ as well as three-nucleon interactions, are also included in the nuclear Hamiltonian. Their successive addition increases the mass limit to 1.80 and 2.20 ${M}_{\ensuremath{\bigodot}}.$ Hamiltonians including a three-nucleon interaction predict a transition in neutron star matter to a phase with neutral pion condensation at a baryon number density of $\ensuremath{\sim}0.2 {\mathrm{fm}}^{\ensuremath{-}3}.$ Neutron stars predicted by these Hamiltonians have a layer with a thickness on the order of tens of meters, over which the density changes rapidly from that of the normal to the condensed phase. The material in this thin layer is a mixture of the two phases. We also investigate 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${M}_{\ensuremath{\bigodot}}$ do not appear to have quark matter admixtures in their cores. However, the heaviest stars are predicted to have cores consisting of a quark and nucleon matter mixture. These admixtures reduce the maximum mass of neutron stars from 2.20 to 2.02 (1.91) ${M}_{\ensuremath{\bigodot}}$ for bag constant $B=200 (122) {\mathrm{M}\mathrm{e}\mathrm{V}/\mathrm{f}\mathrm{m}}^{3}.$ Stars with pure quark matter in their cores are found to be unstable. We also consider the possibility that matter is maximally incompressible above an assumed density, and show that realistic models of nuclear forces limit the maximum mass of neutron stars to be below 2.5${M}_{\ensuremath{\bigodot}}.$ The effects of the phase transitions on the composition of neutron star matter and its adiabatic index $\ensuremath{\Gamma}$ are discussed.