Surface diffusion (SD) is a curvature driven flow where a (hyper-)surface evolves by the surface Laplacian of its mean curvature. It is a fourth order parabolic equation. Compared with its second order counterpart, motion by mean curvature (MMC), for which maximum principle is available, much less is known for SD. I will present a stability result for self-similarity solutions. Though the approach is based on linearization, and it only works for the evolution of graphs, it is quite robust and works for both MMC and SD. I will also present an attempt to analyze the pinch-off phenomena for axisymmetric surfaces. The former is based on a joint work with Hengrong Du and the later with Gavin Glenn.