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The neutrino fast flavor instability (FFI) can change neutrino flavor on time scales of nanoseconds and length scales of centimeters. It is expected to be ubiquitous in core-collapse supernovae and neutron star mergers, potentially modifying the neutrino signal we see, how matter is ejected from these explosions, and the types of heavy elements that form in the ejecta and enrich the universe. There has been a great deal of recent interest in understanding the role the FFI plays in supernovae and mergers, but the short length and time scales and the strong nonlinearity have prevented the FFI from being included consistently in these models. We review the theoretical nature of the FFI starting with the quantum kinetic equations, where the instability exists in neutron star mergers and supernovae, and how the instability behaves after saturation in simplified simulations. We review the proposed methods to test for instability in moment-based calculations where the full distribution is not available and describe the numerical methods used to simulate the instability directly. Finally, we close by outlining the trajectory toward realistic, self-consistent models that will allow a more complete understanding of the impact of the FFI in supernovae and mergers.