A family of spatially third order accurate Compact Upwind Difference schemes is derived for the scalar and vector conservation laws and put in the numerical flux form. A third order flux correction extracted from these schemes is applied to the first order Roe and Osher-Solomon fluxes. A lemma due to Tadmor is then used to limit the high order fluxes of the new schemes so as to render them TVD in the scalar case. Applied to systems, the flux limited versions perform well in limiting the oscillations in the computed solutions.