under CPT, and ones that are odd under CPT. Since in field theory the breaking of CPT symmetry is accompanied by the breaking of Lorentz symmetry, the CPT-breaking terms are necessarily Lorentz breaking. It is reasonable to expect that Lorentz and CPT violation are suppressed at the Planck scale, so the coefficients for Lorentz violation are likely to be small. The interferometric nature of neutrino oscillation experiments, and also of neutral-meson systems, gives them exceptional sensitivity to such tiny effects. This holds promise for oscillation-based experiments to probe new physics and access regions of the SME coefficient space that are still untested. General predictions Current experimental results indicate that neutrinos do indeed oscillate. These oscillations have a variety of possible implications, including the existence of neutrino masses, and the presence of several types of Lorentz violation. In the following, each category of Lorentz breaking is outlined. Spectral anomalies In the standard Lorentz-invariant description of massive-neutrinos, the oscillation phase is proportional to the baseline L and inversely proportional to the neutrino energy E. The mSME introduces dimension-three operators that lead to oscillation phases with no energy dependence. It also introduces dimension-four operators generating oscillation phases proportional to the energy. Standard oscillation amplitudes are controlled by three mixing angles and one phase, all of which are constant. In the SME framework, Lorentz violation can lead to energy-dependent mixing parameters. When the whole SME is considered and nonrenormalizable terms in the theory are not neglected, the energy dependence of the effective hamiltonian takes the form of an infinite series in powers of neutrino energy. The fast growth of elements in the hamiltonian could produce oscillation signals in short-baseline experiment, as in the puma model. The unconventional energy dependence in the theory leads to other novel effects, including corrections to the dispersion relations that would make neutrinos move at velocities other than the speed of light. By this mechanism neutrinos could become faster-than-light particles. The most general form of the neutrino sector of the SME has been constructed by including operators of arbitrary dimension. In this formalism, the speed of propagation of neutrinos is obtained. Some of the interesting new features introduced by the violation of Lorentz invariance include dependence of this velocity on neutrino energy and direction of propagation. Moreover, different neutrino flavors could also have different speeds. L − E conflicts The L − E conflicts refer to null or positive oscillation