Abstract In part I a method is introduced for the numerical solution of swirling axisymmetric laminar flow situations. Consideration is given to finite difference formulation, boundary condition implementation, iterative solution strategies and pressure calculation. Part II is devoted to the application of the method to non-swirling flow problems. Some of these problems are well covered in the literature whilst others are new. Swirling flows are considered in Part III. In particular, details of separation and vortex breakdown induced by swirl are discussed. All results are supplemented by a selection of contour plots to indicate the nature of the flows predicted.