Richtmyer-Meshkov instability and turbulent mixing are fundamental problems of multi-materials interface dynamics, which mainly focuses on the growth of perturbation on the interface and mixing of different materials. It is very important in many applications such as inertial confinement fusion, high-speed combustion, supernova, etc. In this chapter, we will gain advances in understanding this problem by numerical investigations, including the numerical method and program we used, the verification and validation of numerical method and program, the growth laws and mechanics of turbulent mixing, the effects of initial conditions, the dynamic behavior, and some new phenomenon for Richtmyer-Meshkov instability and turbulent mixing.