We examine the presence of strangeness-bearing components, hyperons and kaons, in dense neutron star matter. Calculations are performed using relativistic mean field models, in which both the baryon-baryon and kaon-baryon interactions are mediated by meson exchange. Results of kaon condensation are found to be qualitatively similar to previous work with chiral models, if compatibility of the kaon optical potentials is required. The presence of strangeness, be it in the form of hyperons or kaons, implies a reduction in the maximum mass and a relatively large number of protons, sufficient to allow rapid cooling to take place. The need to improve upon the poorly-known couplings of the strange particles, which determine the composition and structure of neutron stars, is stressed. We also discuss generic problems with effective masses in mean field theories.

36 pages, revtex, figures by fax/mail from authors