Sub-sonic flows in continuum to near continuum regime constitute a major portion of the applied fluid mechanics. Two important examples of such flows are incompressible fluid turbulence and flow through micro-devices. The computational fluid dynamics has not yet realized its full potential for turbulence or flow through micro-devices. While in the case of micro-flows it is the failure of continuum field description by the Navier-Stokes-Fourier equations, for turbulence the difficulty lies in the inability to reduce the description in terms of a few discrete modes.Simulation algorithms based on kinetic theory are well suited to describe the flow in near-continuum as well as continuum regime. Traditionally, two distinct classes of kinetic algorithms are available to perform such simulations. The first class of kinetic simulation algorithms attempts to keep the details of molecular interactions intact and is used to perform purely molecular simulation of the flow. This class of model is computationally too expensive to be used in a realistic simulation in continuum or near continuum sub-sonic regime. The second class of model known as the lattice Boltzmann method tries to construct a bottom-up approach where an extremely simplified kinetic theory is built from the macroscopic equation of motion. This approach is now a well established method for computational fluid dynamics primarily due to a simple and efficient dis cretization scheme. However, such an approach has worked till now, with a reasonable degree of success for incompressible continuum hydrodynamics only.In this thesis an alternate viewpoint, on the minimal kinetic modeling such as the lattice Boltzmann method is presented. Such models are viewed as an efficient temporal integration scheme for model kinetic equations. A class of minimal kinetic models is developed to simulate the sub-sonic hydrodynamics in continuum to near-continuum regime. The hydrodynamic limit for these models is established and a nonlinearly stable extension of the lattice Boltzmann method is proposed to solve these kinetic models in the continuum and near continuum regime.A closed form sub-grid model for hydrodynamics was obtained using the minimal kinetic model. This is the first result which shows the clear advantage of using kinetic models for the construction of coarse-grained hydrodynamic models. The derived sub-grid model and simulation results show that kinetic theory provides a promising alternative approach to the modeling of turbulence.For micro-flows, it is shown that the minimal kinetic models can be used as a reliable simulation tool for the one component gaseous flows. Thus the present minimal kinetic model enlarge the domain of applicability of kinetic solvers of fluid dynamics.