While nuclear reactors may be better known for the free neutrons that they generate, these objects are even more prolific in their production of neutrinos, light uncharged partners of electrons in the Standard Model of particle physics. Current models of neutrino production in nuclear reactors predict detection rates and energy spectra at odds with the existing body of direct reactor neutrino measurements. If these discrepancies are taken seriously, then they indicate a misunderstanding of neutrino production in nuclear reactor cores and/or the fundamental properties of neutrinos. In this talk, I will overview how nuclear reactors work and how they produce neutrinos, and then summarize the theoretical and experimental work that has been recently performed to understand the nature of this anomaly. I will also highlight recent neutrino measurements performed by PROSPECT, which has operated a 4 ton segmented lithium-doped liquid scintillator detector covering baseline ranges of ~7-11 meters from the U235-enriched High Flux Isotope Reactor at Oak Ridge National Laboratory. This experiment has demonstrated the feasibility of precision on-surface reactor antineutrino detection, advanced understanding of antineutrino production by the primary fission isotope U235, and placed world-leading limits on sterile neutrino oscillations.