Abstract: Quadrilateral and hexahedral meshes have irreplaceable advantages in geometry processing, finite element analysis and many other applications because of their excellent element properties and global topology structure. However, there are great challenges to generate such meshes under various requirements raised from the applications, for example low shape approximation error, high cell quality, small number of cells and good topology structure. These requirements often conflict with each other, which makes it hard to find a mesh satisfying all these requirements. In this paper, we systematically introduce the typical recent automatic methods from the view point of manifold parameterization, and analyze their advantages and limitations, then discuss the current major challenges and future works.