We present parametric finite-element approximations of curvature flows for curves in R-d, where d >= 2, as well as for curves on two-dimensional manifolds in R-3. Here we consider the curve shortening flow, the curve diffusion and the elastic flow. It is demonstrated that the curve shortening and the elastic flows on manifolds can be used to compute nontrivial geodesics and that the corresponding geodesic curve diffusion flow leads to solutions of partitioning problems on two-dimensional manifolds in R-3. In addition, we extend these schemes to anisotropic surface energy densities. The presented schemes have very good properties with respect to stability and the distribution of mesh points, and hence no remeshing is needed in practice.