: In this thesis, we show that the Lorentz violation effects in the Standard Model Extension (SME) can produce circular polarization in CMB photons. By computing the Compton scattering of CMB radiation in the Lorentz violated effects, we found some new terms that changed the time evolution of the stokes parameter V. We consider the Lorentz violated extension of the standard model. In this framework there are some terms that explicitly violate the CP-symmetry. We examine the CPT-eve -term to find the electron electric dipole moment. We have shown that the form factors besides the momentum transfer, depend on new Lorentz- scalars constructing by and the four momenta of the electron as well. Such an energy dependence of the electric dipole form factor leads to an enhancement of the electron electric dipole moment at high energy even at the zero momentum transfer. We show that at the electric dipole moment of the electron can be as large as ecm . Furthermore, we considere the - term in the SME lagrangian and we compute the contribution of this term in the g-factor of muon. We have shown that the form factors besides the momentum transfer, depend on a new Lorentz- scalar constructing by and the four momenta of the muon as well. By using the discrepancy between the standard model prediction and the measurement of the anomalous magnetic moment for muon, we put a bound of the order of for the muon Lorentz violation coefficient . Due to the quantum gravity fluctuations at the Planck scale, the space-time manifold is no longer continuous, but discretized. As a result the Lorentz symmetry is broken at very high energies. We have studied the neutrino oscillation pattern due to the Lorentz Invariance Violation (LIV), and compare it with the normal neutrino oscillation pattern due to neutrino masses. We have found that at very high energies, neutrino oscillation pattern is very different from the normal one. This could provide an possibility to study the Lorentz Invariance Violation by measuring the oscillation pattern of the very high energy neutrinos from a cosmological distance. Key Words : Lorentz symmetry violation, Standard Model Extension, Circular polarization of the CMB radiation, Electric dipole moment, Muon’s g-factor, High energy Neutrino oscillations