Abstract: The Willmore energy of a two-dimensional surface is given by the integral of the mean curvature squared. It is a type of bending energy and a special case of the Helfrich energy, used to describe lipid bilayers. An important feature is its invariance with respect to smooth conformal transformations. In this talk we consider the steepest descent flow associated to this energy and we concentrare to the case that the initial datum is a torus of revolution. We discuss two variants of this flow and present under which conditions long-time existence results can be achieved. We also plan to discuss some new insights on the asymptotic behavior of the solutions. This is joint work with M. Müller (U Augsburg), F. Rupp (U Vienna) and M. Schlierf (U Ulm).