An attempt has been made to reconcile and to critically analyze the voluminous literature available on the Row of theologically complex fluids through unconsolidated fixed beds and fluidised beds. In particular, consideration is given to the prediction of macro-scale phenomena of flow regimes, pressure drop in fixed and fluidised beds, minimum fluidisation velocity, dispersion and liquid-solid mass transfer. Available scant results seem to suggest that flow patterns qualitatively similar to that observed for Newtonian fluids, can be expected for the flow of purely viscous non-Newtonian fluids. A Reynolds number based on the effective pore size and pore velocity is seen to be a convenient parameter for the delineation of these flow regimes. Out of the four approaches available, the generalisation of the capillary model, due to Comiti and Renaud (Chem. Engng. Sci. 44 (1989) 1539-1545), appears to be the best for the estimation of the pressure drop through fixed beds. This method requires the flow rate - pressure drop data for the Row of a Newtonian fluid, such as air or water, through the same bed to evaluate the two key parameters, namely, the tortuosity and the dynamic surface area. While this approach can accommodate non-spherical particle shape and the wall effects and encompasses all possible flow regimes, it is limited to the situations where the polymer-wall interactions are negligible, Similarly, based on a combination of the capillary and drag models, satisfactory expressions have been identified for the prediction of the minimum fluidising velocity and velocity-voidage behaviour of uniformly expanded fluidised beds for power-law liquids and beds of spherical particles, Little is known about the effect of particle shape and column walls on these parameters. Even less work has been reported on dispersion and liquid-solid mass transfer in packed and fluidised beds, and no theoretical or experimental results seem to be available on heat transfer in these systems. Therefore, the expressions for the prediction of Peclet and Sherwood numbers presented herein must be regarded as somewhat tentative at this stage. Finally, little definitive and quantitative information is available on the role of viscoelasticity and of the effects arising from polymer/wall interactions, polymer retention, etc.