The application of total variation diminishing (TVD) implicit and explicit schemes to transonic flow around airfoils is presented. Several upwind and symmetric TVD flux limiters from the Non-MUSCL family are examined and compared with Jameson's non-linear artificial dissipation model. Inviscid results indicate that TVD schemes reduce shock smearing while practically eliminating numerical oscillations. Analytical results are presented for a NACA 0012 airfoil for various transonic flow conditions using TVD formulations of the 2D Navier-Stokes equations. Comparisons of computed results with experimental data show good agreement.