Lorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance. Today, neutrino oscillation or change of one type of neutrino into another is an experimentally verified fact; however, the details of the underlying theory responsible for these processes remain an open issue and an active field of study. The conventional model of neutrino oscillations assumes that neutrinos are massive, which provides a successful description of a wide variety of experiments; however, there are a few oscillation signals that cannot be accommodated within this model, which motivates the study of other descriptions. In a theory with Lorentz violation, neutrinos can oscillate with and without masses and many other novel effects described below appear. The generalization of the theory by incorporating Lorentz violation has shown to provide alternative scenarios to explain all the established experimental data through the construction of global models. Introduction Conventional Lorentz-preserving descriptions of neutrinos explain the phenomenon of oscillations by endowing these particles with mass. However, if Lorentz violation occurs, oscillations could be due to other mechanisms. The general framework for Lorentz violation is called the Standard-Model Extension (SME). The neutrino sector of the SME provides a description of how Lorentz and CPT violation would affect neutrino propagation, interactions, and oscillations. This neutrino framework first appeared in 1997 as part of the general SME for Lorentz violation in particle physics, which is built from the operators of the Standard Model. An isotropic limit of the SME, including a discussion on Lorentz-violating neutrino oscillations, was presented in a 1999 publication. Full details of the general formalism for Lorentz and CPT symmetry in the neutrino sector appeared in a 2004 publication. This work presented the minimal SME (mSME) for the neutrino sector, which involves only renormalizable terms. The incorporation of operators of arbitrary dimension in the neutrino sector was presented in 2011. The Lorentz-violating contributions to the Lagrangian are built as observer Lorentz scalars by contracting standard field operators with controlling quantities called coefficients for Lorentz violation. These coefficients, arising from the spontaneous breaking of Lorentz symmetry, lead to non-standard effects that could be observed in current experiments. Tests of Lorentz symmetry attempt to measure these coefficients. A nonzero result would indicate Lorentz violation. The construction of the neutrino sector of the SME includes the Lorentz-invariant terms of the standard neutrino massive model, Lorentz-violating terms that are even