Perfect, partial, and approximate symmetries are pervasive in 3D surface meshes of real-world objects. However, current digital geometry processing algorithms generally ignore them, instead focusing on local shape features and differential surface properties. This paper investigates how detection of large-scale symmetries can be used to guide processing of 3D meshes. It investigates a framework for mesh processing that includes steps for symmetrization (applying a warp to make a surface more symmetric) and symmetric remeshing (approximating a surface with a mesh having symmetric topology). These steps can be used to enhance the symmetries of a mesh, to decompose a mesh into its symmetric parts and asymmetric residuals, and to establish correspondences between symmetric mesh features. Applications are demonstrated for modeling, beautification, and simplification of nearly symmetric surfaces.