Lorentz violation and supersymmetry

We review recent progress in constructing supersymmetric field theories modified by the presence of constant vector and tensor backgrounds, or in other words SUSY field theories with Lorentz violation (LV). We show that the requirement of supersymmetry and gauge invariance limits the LV in the minimal supersymmetric Standard Model to dimension five and higher levels, giving a natural explanation to the smallness of the Lorentz violation. We show main phenomenological features of such models, among which is the absence of large LV corrections to dispersion relations, and include radiative corrections. We show that 1/M_planck normalized corrections are ruled out at the 10 ^-10 level.