Dispersive PDEs, such as Klein-Gordon equation, Dirac equation, Schrodinger equation, arise from many different areas, e.g. computational chemistry, plasma physics, quantum mechanics. Typical computational tasks in dispersive PDEs are finding the ground/stationary states and solving the dynamics. In this talk, we report some recent advances on the numerical methods and analysis for the time-dependent dispersive PDEs, paying particular attention to the highly oscillatory PDEs, which usually exhibit solutions with high frequency waves in time and/or in space, and are generally computational expensive.