The authors report on a calculation completion for the numerical conformal mapping of the exterior of the unit disc to the exterior of a smooth curve. The competitors are methods of Theodorsen, Timman, Friberg and the reviewer, which have in common that they all make use of function conjugation. After a short description of these methods results are presented for six classes of examples: ellipses (with a very simple mapping function), sport-grounds, inverted ellipses (difficult in view of the crowding phenomenon), a cosine airfoil (pretreated by a Karman- Trefftz map), perturbed circles and general spline curves. Several nice figures show the conformal maps for different parameters. Other figures show the minimal errors achieved by the different methods with a number N of grid points. These errors decrease exponentially for analytic boundary curves, but like a power of N for k-times differentiable boundary curves such as e.g. spline curves. This confirms theoretical predictions which estimate the discretization error by the truncation error of Fourier series. This simple picture is complicated by the truncation error of Fourier series. This simple picture is complicated by the observation that the methods of Timman, Friberg and the reviewer in some cases seem to converge initially but diverge finally. For this phenomenon see also the reviewer, J. Comput. Appl. Math. 29, No.2, 207-244 (1990; Zbl 0695.30005). The authors’ conclusion: “... Wegmann is the most efficient and most robust”. The reviewer’s conclusion: This paper contains a wealth of experimental material which is useful and stimulating for the user as well as for the theoretician.