It is generally agreed on that the tremendous densities reached in the centers of neutron stars provide a high-pressure environment in which several intriguing particles processes may compete with each other. These range from the generation of hyperons to quark deconfinement to the formation of kaon condensates and H-matter. There are theoretical suggestions of even more exotic processes inside neutron stars, such as the formation of absolutely stable strange quark matter. In the latter event, neutron stars would be largely composed of strange quark matter possibly enveloped in a thin nuclear crust. This paper gives a brief overview of these striking physical possibilities with an emphasis on the role played by strangeness in neutron star matter, which constitutes compressed baryonic matter at ultra-high baryon number density but low temperature which is not accessible to relativistic heavy ion collision experiments.