||In this thesis, the gas-kinetic BGK scheme for two-dimensional viscous flow compu-tation is extended to the moving meshes. Specifically, both the static adaptive grid method and dynamic unified moving mesh method have been developed. In the former one, the grid movement and adaptation are controlled by a monitor function which depends on the gradient of flow variables, such as density or velocity. The grid points in the adaptive grid method can be easily moved and concentrated to the regions with large density and velocity gradients, such as multi-material interface and the bound-ary layer. Therefore, the adaptive grid method is more accurate and efficient than the methods with stationary mesh points. In the dynamic moving mesh method, the gas-kinetic BGK equation is first reformulated under a unified coordinate transforma-tion with grid velocity included. Then, a unified conservative gas-kinetic scheme is constructed for the viscous flow computation on moving meshes. Due to the coupling between the grid velocity and the overall solution algorithm, the Eulerian and La-grangian methods become two limiting cases in the current gas-kinetic method. The moving grid method extends the applicable regime of the gas-kinetic scheme to the flows with free surface and moving boundaries, such as dam break problem and airfoil oscillations. In order to further increase the robustness of the moving grid scheme, the above two methods, i.e., static and dynamic ones, have been uniquely combined to move, redistribute, and remedy the distorted meshes in the fluid computations. Many numerical examples from incompressible low speed flow to the supersonic shock interaction are presented. The accuracy and robustness of the moving mesh have been fully demonstrated. In the unsteady aerodynamic flow application, the falling plate problems with the rich dynamic behavior, such as tumbling and fluttering, have been studied. Excellent agreement between the experimental measurements and the numerical computations has been obtained.