This short review aims at giving a brief overview of various states of matter that have been suggested to exist in the ultra-dense centers of neutron stars. Particular emphasis is put on the role of quark deconfinement in neutron stars and on the possible existence of compact stars made of absolutely stable strange quark matter (strange stars). Astrophysical phenomena, which distinguish neutron stars from quark stars, are discussed and the question of whether or not quark deconfinement may occur in neutron stars is investigated. Combined with observed astrophysical data, such studies are invaluable to delineate the complex structure of compressed baryonic matter and to put firm constraints on the largely unknown equation of state of such matter.Instituto de Física La PlataFacultad de Ciencias Astronómicas y Geofísicas