The curvature-driven flows play an important role in applied mathematics and materials science. One of the most prominent examples is the surface diffusion flow, and the normal velocity of the interface is given by the surface Laplacian of the mean curvature. In this talk I will present structure-preserving parametric finite element methods (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D), a surface in three dimensions (3D). Here the “structure-preserving” refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. The proposed schemes are based on weak formulations that allow tangential degrees of freedom and thus lead to discretizations with good mesh quality. The exact area/volume conservation is maintained with the help of suitably weighted discrete normals. The generalization of the SPFEM to the axisymmetric surface diffusion, the anisotropic surface diffusion for surfaces attached to external plane boundaries will also be considered.