Superconvergence phenomenon is well understood for theh-version finite element method and researchers in this old field have accumulated a vast literature during the past 40 years. However, the relevant systematic study for discontinuous Galerkin, finite volume, and spectral methods is lacking. We believe that the scientific community would also benefit from the study of superconvergence phenomenon of those methods. Recently, some efforts have been made to expand the territory of the superconvergence. In this talk, I will summarize some recent development on superconvergence study for these methods. At the same time, some current issues and un-solved problems will also be addressed.