The thesis is concerned with a theoretical study of the flow behaviour of inelastic power law fluids in two different types of flow situation. These are: 1. The creeping motion of a sphere moving through an expanse of liquid. 2. The combined steady and oscillatory flow of a liquid through a straight tube of circular cross section. The first part of the work is devoted to the prediction of the drag correction factor for a sphere falling slowly through a bounded inelastic power law fluid. The analysis is carried out for the case when the outer spherical boundary has a finite or infinite radius. A perturbation technique is used to produce the resulting equations for a slightly power law fluid which are solved using the finite element method. An asymptotic expansion is used to provide an analytical far field solution for the infinite outer sphere case. The second part considers the combined steady and oscillatory flow of an inelastic power law liquid in a tube. The analysis is carried. out for the case when both the steady flow rate and the oscillatory flow rate are known. An expression for the pressure gradient reduction in the tube is then derived. The resulting partial differential equation is solved by finite difference techniques. An analytical solution for the pressure gradient is also obtained using a perturbation analysis for the case when the fluid inertial effects are small.