||This thesis studies numerical methods for a mixed Stokes-Darcy problem, which models the coupling of surface and subsurface flows. We are interested in developing efficient decoupling methods for this coupled problem for a number of appealing reasons. We first describe a two-grid method for the stationary mixed Stokes-Darcy problem. The basic idea of the method is to first solve a much smaller problem on a coarse grid, and then use the coarse grid solution to interpolate the interface coupling conditions, which leads to a decoupled problem on a fine grid. Numerical experiments are carried out to show the effectiveness of this method. Note that the key idea of the two-grid decoupling approach for the stationary model is to approximate the interface coupling term by a coarse-grid solution. We may directly extend the above two-grid approach to the non-stationary case by applying it at each time level using an implicit marching scheme in the time discretization. This, however, requires solving a coupled coarse-grid problem at each time level. Numerical experiments are reported. Furthermore, by taking advantage of the nature of time-dependent problems that the computed solutions from previous time levels can provide useful information for various approximation purposes via suitable temporal extrapolation techniques, we present a fully decoupled marching scheme by interface approximation with temporal extrapolation for the non-stationary mixed Stokes-Darcy problem. Stability and convergence analyses are presented. Numerical experiments are also conducted to demonstrate the computational effectiveness of this decoupling approach.