The cores of neutron stars have nonrelativistic matter beyond nuclear saturation density, making them valuable complements to terrestrial laboratory experiments in the study of dense matter. However, their remoteness means that we can only observe them rather than experiment on them, and this leads to potential systematic uncertainties. I will discuss multiple ways that neutron star observations have been used to constrain the properties of high-density matter, and will in each case give a candid appraisal of the current reliability of the methods as well as future prospects. Examples include binary mass measurements, radius estimates from thermal spectra, constraints from modeling of thermonuclear burst oscillations, and information from cooling of transient accretors.