Interstellar hydrides -that is, molecules containing a single heavy element atom with one or more hydrogen atoms -were among the first molecules detected outside the solar system. They lie at the root of interstellar chemistry, being among the first species to form in initially-atomic gas, along with molecular hydrogen and its associated ions. Because the chemical pathways leading to the formation of interstellar hydrides are relatively simple, the analysis of the observed abundances is relatively straightforward and provides key information about the environments where hydrides are found. Recent years have seen rapid progress in our understanding of interstellar hydrides, thanks largely to far-IR and submillimeter observations performed with the Herschel Space Observatory. In this review, we will discuss observations of interstellar hydrides, along with the advanced modeling approaches that have been used to interpret them, and the unique information that has thereby been obtained.