A multiple-step pressure correction algorithm, similar in spirit to the PISO algorithm, has been developed for the calculation of viscous flows in nonorthogonal curvilinear coordinates. The resulting pressure correction equations arc solved using a multigrid correction scheme. These developments were pursued because the pressure correction equation is the most time-consuming part of a Navier-Stokes flow calculation and because of the effect of the coupling between the pressure and velocity variables on the convergence rate. The new algorithm does improve the convergence rate for laminar flows or flows calculated on nearly orthogonal meshes. However, for turbulent or reacting flows, or flows computed on highly nonorthogonal meshes, there is little or no improvement in the convergence rate, and the CPU time is generally higher than for a single-step algorithm. A suitable balance between updating the velocity and static pressure variables is important, since a tightly converged pressure field can exaggerate the changes in the velocity field and cause the overall convergence rate to worsen. Although the multigrid scheme does a much better job of solving the pressure correction equation to high accuracy than a single-grid method, this limits the improvement of the overall algorithm that is possible. These results indicate that a multiple-step pressure correction algorithm does not have a decisive advantage over a single-step algorithm for many practical flows.