An implicit, space-marching, finite-difference procedure is presented for solving the primitive variable form of the steady, compressible, Navier-Stokes equations in body-fitted, curvilinear coordinates. The numerical procedure uses the primitive variables of velocity and pressure and is suitable for simulating both incompressible and compressible flow. The algorithm employs multiple sweeps of the space-marching procedure to solve a coupled system of equations consisting of the Cartesian coordinate momentum equations and the continuity equation. A new pressure correction method is described and is shown to significantly accelerate the convergence rate;The procedure is used to simulate incompressible and subsonic adiabatic flows over a broad range of Reynolds numbers in the laminar flow regime. Computed results are compared with other numerical predictions for the developing flow in two-dimensional and three-dimensional channel inlets, the separated flow in two-dimensional channels with sudden expansions or contractions, and the crossflow over a cylinder. The method is shown to predict low Reynolds number flows more accurately than schemes based upon approximations of the Navier-Stokes equations;Results of initial studies to optimize the performance of the global pressure correction procedure are also presented and discussed. The new scheme is effective in rapidly propagating the downstream pressure boundary conditions throughout the flow field. Computed results for simulations of internal flows without large obstructions indicate that the convergence rate is not strongly dependent on the number of nodes.