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), and an axisymmetric surface. 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 proper discretizations of weak formulations that allow tangential degrees of freedom. The exact area/volume conservation is maintained with the help of a vector approximation which combines the information of current and next time steps. The methods are implicit and the resulting nonlinear system can be solved via Newton ’ s method. Numerical results will be presented to demonstrate the accuracy and efficiency for computing the surface diffusion flow.