While nuclear reactors may be better known for the neutrons they produce, 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. New reactor antineutrino measurements performed at short distances from highly-enriched uranium reactors will enable independent testing of both of possible explanations for these reactor neutrino anomalies. In this talk, I will overview how nuclear reactors work and how they produce neutrinos, and then summarize the theoretical and experimental work the Littlejohn research group has recently performed to establish the existence and nature of this anomaly. I will also talk about what studying this anomaly has taught us -- from a better understanding how nuclear reactors make heat to new tests for physics beyond the Standard Model.