In this thesis, we give a brief review on the basic concepts of symmetries and conservation laws, in particular, the Lorentz and ymmetries. Lorentz andymmetries are the exact symmetries of nature. However, the breakdown of the Lorentz invariance naturally arises in different scenarios of physics at the Planck scale. Therefore, the expected energy scale in which a Lorentz-violating effect can be happened enormously is about the Planck mass. Meanwhile, the Lorentz andymmetry-violation can be incorporated into existing theories by spontaneous symmetry-breaking. Consequently, in the next step we consider the spontaneous symmetry breaking of these symmetries. Then, we introduce two kinds of the Lorentz transformation: observer and particle Lorentz transformation. It turns out that when we speak of the Lorentz symmetry breaking, we mean a theory or model in which the observer is independent of transformation but is not invariant under particle Lorentz translation. Then we introduce the Standard Model Extension. This model provides the most general observer-independent field theoretical framework for investigations of Lorentz violation. Subsequently, a general Lorentz-breaking extension of quantum electrodynamics is extracted from the , and some experimental tests in this framework are considered. Then we explore the electromagnetic form factors in the Standard Model and beyond SM. We show that in models beyondthat preserves Lorentz symmetry, in addition to the electric charge and magnetic dipole form factors, the electric dipole and anapole form factors are also appeared. Then we examine the form factors in the . In this framework the electric dipole form factor besides the momentum transfer, depend on a new Lorentz-scalar, constructing by Lorentz violating parameter, and the four momenta of the lepton, as well. Such an energy dependence of the electric dipole form factor leads to an enhancement of the lepton electric dipole moment at high energy, even at the zero momentum transfer. Then assuming interaction of the electric dipole moment ofandleptons with photon and their weak dipole moments with the oson, we calculate the total cross-section of the reaction Consequently, we drive the upper limits on the Lorentz violating parameter, forand leptons by using the obtained value of the electric dipole moment in the . This limit is about for and for .