In this talk I shall discuss some recent progresses in developing interior penalty discontinuous Galerkin (IPDG) methods and local discontinuous Galerkin (LDG) methods for high frequency scalar wave equation. The focus of the talk is to present some non-standard (h- and hp-) IPDG and LDG methods which are proved to be absolutely stable (with respect to the wave number and the mesh size) and optimally convergent (with respect to the mesh size). The proposed IPDG and LDG methods are shown to be superior over standard finite element and finite difference methods, which are known only to be stable under some stringent mesh constraints. In particular, it is observed that these non-standard IPDG and LDG methods are capable to correctly track the phases of the highly oscillatory waves even when the mesh violates the "rule-of-thumb" condition. Numerical experiments will be presented to show the efficiency of the non-standard IPDG and LDG methods. If time permits, latest generalizations of these DG methods to the high frequency Maxwell equations will also be discussed. This is a joint work with Haijun Wu of Nanjing University (China) and Yulong Xing of the University of Tennessee and Oak Ridge National Laboratory.