The two-dimensional depth-averaged shallow water equations have attracted considerable attentions as a practical way to solve flows with free surface. Compared to three-dimensional Navier-Stokes equations, the shallow water equations give essentially the same results at much lower cost. Solving the shallow water equations by the Godunov-type finite volume method is a newly emerging area. The Godunov-type finite volume method is good at capturing the discontinuous fronts in numerical solutions. This makes the method suitable for solving the system of shallow water equations. In this dissertation, both the shallow water equations and the Godunov-type finite volume method are described in detail. A new surface flow routing method is proposed in the dissertation. The method does not limit the shallow water equations to open channels but extends the shallow water equations to the whole domain. Results show that the new routing method is a promising method for prediction of watershed runoff. The method is also applied to turbulence modeling of free surface flow. The κ - ε turbulence model is incorporated into the system of shallow water equations. The outcomes prove that the turbulence modeling is necessary for calculation of free surface flow. At last part of the dissertation, a total load sediment transport model is described and the model is tested against 1D and 2D laboratory experiments. In summary, the proposed numerical method shows good potential in solving free surface flow problems. And future development will be focusing on river meandering simulation, non-equilibrium sediment transport and surface flow - subsurface flow interaction.