The high-density nuclear equation of state within the Brueckner-Hartree-Fock approach is discussed. Particular attention is paid to the effects of nucleonic three-body forces, the presence of hyperons, and the joining with an eventual quark matter phase. The resulting properties of neutron stars, in particular the mass-radius relation, are determined. It turns out that in our approach the maximum mass is about 1.7 solar masses and that stars heavier than about 1.3 solar masses contain necessarily quark matter.