In this thesis, at first we introduce Lorentz and CPT symmetries that are exact symmetries of nature and then we explain about Lorentz symmetry violation as a signature of Planck-Scale. We introduce a suitable frame for spontaneous symmetry breaking of Lorentz violation that is called Standard Model Extension (SME). The Standard Model Extension provides a quantitative description of the Lorentz and CPT violation that intern is controlled by a set of coefficients whose the values are to be constrained by experiments. Einstein's special relativity violation is explained in the quantum electrodynamics sector of SME. We focus on this sector and study the experiments which are performed in this area. Consequently we investigate non-commutative theory including Lorentz Violation with Weyl-Moyal formalism and its capabilities. The theory that is constructed only with Weyl-Moyal formalism suffers from some major problems that we review some of them. Hence non-commutative theory can be constructed in two different approaches. In the first one, the non-commutative fields are the same as the ordinary ones while the gauge group is restricted to U(n). For example the symmetry group of the standard model in the non-commutative space is U(3)×(2)×U(1) which can be reduced to SU(3)×SU(2)×U(1) by two appropriate spontaneous symmetry breaking. In contrast, in the second approach the non-commutative gauge theory can be constructed for SU(n) gauge group via Seiberg-Witten map. Since the non-commutative theory has Lorentz violation, it can be considered as a subset of the Standard Model Extension. Subsequently we derive and determine the relation of the non-commutative parameter with the Lorentz violation parameters for the two approaches and compare the obtained results. For example in the first approach there are a µ , c µ? and k F Lorentz violation coefficients while only c µ? and k F are nonzero in the second one. At the end, we obtain new limits on the non-commutative parameter by using the existing bounds on the Lorentz Violation parameters. For instance by using the existing bounds on the Lorentz violation coefficient (c), we find a bound as high as 22 TeV on the non-commutative parameter. Keywords: Lorentz symmetry violation, Standard Model Extension, Non-Commutative theory, Seiberg-Witten map