  Properties of dense nucleon matter and the structure of neutron stars are
studied using variational chain summation methods and the new Argonne v18
two-nucleon interaction. The neutron star gravitational mass limit obtained
with this interaction is 1.67 M_{solar}. 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_{solar}. 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 0.2 fm^{-3}. 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_{solar} 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_{solar} for bag constant B = 200 (122) MeV/fm^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_{solar}. The effects of the phase transitions on the composition
of neutron star matter and its adiabatic index are discussed.
