There is audio interference. We start with a brief introduction to the conforming finite element method on polygonal/polyhedral meshes using the generalized barycentric coordinates. Then we present our recent work in this direction on the construction of a H(div)/H(curl) conforming element and a Crouzeix-Raviart type non-conforming element. Finally, we discuss how to develop an alignment and an equidistribution quality measures for polytopal meshes, and how to build a Moving Mesh PDE (MMPDE) moving mesh method based on these mesh quality measures. The work on H(div)/H(curl) conforming element is done in collaboration with Dr. Wenbin Chen. The work on mesh quality and moving mesh method is done in collaboration with Dr. Weizhang Huang.