For over 40 years, physicists have considered possible uses for neutrino detectors in nuclear nonproliferation, arms control, and fissile materials security. Neutrinos are an attractive fission signature because they readily pass through matter. The same property makes neutrinos challenging to detect in systems that would be practical for nuclear security applications. This Colloquium presents a broad overview of several potential neutrino applications, including the near-field monitoring of known reactors, far-field monitoring of known or discovery of undeclared reactors, detection of reactor waste streams, and detection of nuclear explosions. Recent detector advances have made near-field monitoring feasible, whereas farther-field reactor detection and waste stream detection monitoring may be possible in some cases with further research and development. Very long-range reactor monitoring and nuclear explosion detection do not appear feasible for the foreseeable future due to considerable physical and/or practical constraints.