The state-of-art methods for simulating high frequency wave propagation rely on the high frequency theories, i.e., geometrical optics.  The waves  before caustics appear can be constructed accurately by geometrical optics if source singularities for the amplitude and phase can be resolved.  In this talk, we introduce systematic approaches to resolve the source singularities of amplitude and phase. More importantly, we introduce a new method  to simulate
the highly oscillating waves beyond geometrical optics even when caustics appear. Numerical examples will be presented to demonstrate the methods. (with Profs. Robert Burridge, Jianliang Qian and Hongkai Zhao).