The merger of a binary composed of a neutron star and/or a black hole is one of the most promising source of gravitational waves. If we detected gravitational waves from them, it could tell us a validity of the general relativity in a strong gravitational field and the equation of state of neutron star matter. Furthermore, if gravitational waves from a compact binary merger and a short-hard gamma- ray burst are observed simultaneously, a long-standing puzzle on the central engine of short gamma-ray bursts could be resolved. In addition, compact binary mergers are a theoretical candidate of the rapid process nucleosynthesis site. Motivated by these facts, it is mandatory to build a physically reliable model of compact binary mergers and numerical relativity is a unique approach for this purpose. We are tackling this problem from several directions; the magneto-hydrodynamics, the neutrino radiation transfer, and a comprehensive study with simplified models. I will talk our understanding and future prospect on the compact binary mergers.