||The gas-kinetic Bhatnagar-Gross-Krook (BGK) finite volume method has been successfully proposed and developed for both inviscid and viscous flow simulations in the past decade. In this thesis, the gas-kinetic method is extended in terms of three aspects: the gas-kinetic Runge-Kutta discontinuous Galerkin (RKDG) finite element method for viscous flow equations was successfully constructed and numerically tested; a multiscale gas-kinetic approach was proposed and applied for continuum and near continuum flow simulations; and a multiple translational temperature kinetic model was presented for microscale flow problems. In recent years, high-order numerical methods have been extensively investigated and widely used in computational fluid dynamics (CFD), in order to effectively resolve complex flow features using meshes which are reasonable for today’s computers. Here we refer to high-order methods by those with order of accuracy at least three. Among the most successful high-order formulations is the discontinuous Galerkin (DG) finite element method. In the first part of this thesis, a RKDG method based on the gaskinetic formulation for viscous flow computation is proposed, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gaskinetic BGK model and the Navier-Stokes (NS) equations, the NS flux is automatically obtained by the present method. Numerical examples for both one-dimensional (1D) and two-dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method. In the near continuum regime, both solid mathematical models and efficient numerical algorithms are highly desired. In the second part of this thesis, we investigate the performance of the gas-kinetic approach with the generalized particle collision time, which depends on both first- and second-order derivatives of the local flow variables, for continuum and near continuum flow problems. Based on this model, the nonequilibrium shock structure, Poiseuille flow, nonlinear heat conduction problems, and the unsteady Rayleigh problem will be calculated. The numerical results are in good agreement with those by both the direct simulation Monte Carlo (DSMC) method and experience. Microscale gas flow is a new rapidly growing research field being driven by microsystems technology. It has been shown that the fluid mechanics of microscale gas flows are not the same as those experienced in the macroscopic world. In the third part of the thesis, a gas-kinetic model with multiple translational temperature for monatomic gas is proposed for microscale flow simulations. In the continuum regime, the standard NS solutions are precisely recovered. For the microscale gas flows, the simulation results are compared with both DSMC data and solutions of linearized Boltzmann equation. Numerical experiments demonstrate that the multiple temperature kinetic model has the advantage over the standard Navier-Stokes equations in capturing the non-equilibrium physical phenomena for the microscale gas flow simulation. It is clearly shown that many thermal nonequilibrium phenomena in microscale flows can be well captured by modifying some assumptions in the standard NS equations.