In this talk, I begin with a review of different geometric flows (PDEs) including mean curvature (curve shortening) flow,surface diffusion flow, Willmore flow, etc., which arise from materials science, interface dynamics in multi-phase flows,biology membrane, computer graphics, geometry, etc. Different mathematical formulations and numerical methods for mean curvature flow are then discussed. In particular,an energy-stable linearly implicit parametric finite element method(PFEM) is presented in details. Then the PFEM is extended to surface diffusion flow and anisotropic surface diffusion flow,and a structure-preserving implicit PFEM is proposed.Finally, sharp interface models and their PFEM approximations are presented for solid-state dewetting. This talk is based on joint works with Harald Garcke, Wei Jiang, Yifei Li,Robert Nuernberg, Yan Wang and Quan Zhao.