An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two-dimensional incompressible Navier-Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non-staggered grid arrangement. The problem of pressure-velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second-order accuracy for both diffusion- and convection-dominated flows. There is little loss of accuracy on grids that are moderately non-orthogonal. An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two-dimensional incompressible Navier-Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non-staggered grid arrangement. The problem of pressure-velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second-order accuracy for both diffusion- and convection-dominated flows. There is little loss of accuracy on grids that are moderately non-orthogonal.