Abstract: We present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in curvature space -- we develop the precise conditions under which curvature is allowed to evolve. The practical outcome is a highly efficient algorithm that naturally preserves texture and does not require remeshing during the flow. We apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces. We also present a new algorithm for length-preserving flow on planar curves, which provides a valuable analogy for the surface case.