This thesis presents the results of the investigations in two areas, i.e. non-Newtonian annular flow and cuttings transport in drilling annuli at various angles. In the first part of the thesis, a review of the fundamentals and the previous studies on laminar concentric annular flow of non-Newtonian fluids is given at first. Then two parallel theoretical studies are performed, respectively, on: a. Laminar eccentric annular flow of power-law and Bingham plastic fluids. In this analysis, a new method is used which treats an eccentric annulus as infinite number of concentric annuli with variable outer radius. The analytical solutions of the shear stress, shear rate, velocity and the volumetric flowrate/pressure gradient are obtained over the entire eccentric annulus. This analysis is useful in design of any engineering operations related to eccentric annular flow such as oil drilling operations. b. Laminar helical flow of power-law fluids through concentric annuli. A group of dimensionless equations are derived in this analysis for the profiles of the apparent viscosity, angular and .axial velocities, and for the volumetric flowrate. These equations are essential when one needs to simulate the helical flow conditions in various engineering operations. In addition, another group of dimensionless equations are also derived for pressure gradient calculations which can be used directly by drilling engineers to predict the reduction of the annular friction pressure drop caused by drillpipe rotation during drilling operations. The second part of the thesis is dedicated to the investigations into the problems directly related to cuttings transport through drilling annuli at various angles. First, both theoretical and experimental studies on settling velocities of drilled cuttings in drilling fluids are conducted using new approaches to account for the non-Newtonian nature of drilling fluids and for the shape irregularity of drilled cuttings. Based on experimental results, a generalised model is developed for calculating settling velocities of variously shaped particles in power-law fluids. Then, the effects of various parameters on cuttings transport during drilling operations are analysed based on the previous and the present studies. After that, an extensive theoretical analysis for the previous studies on the minimum transport velocity (MTV) in solid-liquid mixture flow through pipelines, on initiation of sediment transport in open channels and on MTV for cuttings transport in deviated wells is presented. At last, theoretical studies on the minimum transport velocity for cuttings transport in drilling annuli at various angles are conducted and two parallel general correlations are developed. When these correlations are experimentally verified and numerically established in the future, they can be served as general criteria for evaluating and correlating the effects of various parameters on cuttings transport, and as a guideline for cuttings transport programme design during directional drilling.