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There are hints coming from some scenarios of modern String and Quantum Gravity theories that Lorentz invariance may not be an exact symmetry of Nature. The study of Lorentz violating theories therefore provides an insight into ultraviolet physics. We employ the Effective Field Theory technique to study the most general extension of the Standard Model and its Supersymmetric modifications with Lorentz-violating interactions of mass dimension five. We provide a complete classification of the interactions in these theories and determine a typical experimental sensitivity to the size of Lorentz violation. A detailed study of the operators that induce CPT-odd Electric Dipole Moments is performed and it is shown that they yield an independent constraint on Lorentz violating physics. We provide an application of Lorentz violating physics to the problem of generation of baryon asymmetry of the universe. A scenario of Leptogenesis driven by CPT-odd interactions is considered and confronted with experimental constraints on Lorentz-violating physics.