This thesis explores a number of hydrodynamic sedimentation problems in which the fluid motion is governed by the quasi-steady Stokes approximation. In Chapter Two potential-theoretic methods are used to solve the axisymmetric sedimentation problem of a small slowly rotating and translating particle moving through a fluid-filled circular pore of finite length. In such an approach the velocity and pressure fields, governing the fluid motion, are expressed in terms of relevant harmonic functions, a method which is also applicable to linear elasticity. To give the motivation behind the selection of these harmonics and the analytic methods used, Chapter One discusses two electrostatic problems possessing analogous boundary conditions. In both chapters each problem is reduced to the solution of coupled infinite systems of linear equations and solved by truncation to coupled finite sets. The numerical solutions of these equations are then used to compute approximations to the resistive torque and drag experienced by the sedimenting particle. Chapter Three is divided into two parts. Part I treats the application of the method of subareas to some model electrostatic problems. In Part II we utilize the Oseen integral representation of solutions of the Stokes equations to develop a boundary-integral method for the study of further sedimentation problems. The method involves the discretization of the linear integral equations of the first kind, whose unknowns comprise boundary-stress components, using the method of subareas, thus reducing the integral equations to a system of linear equations. Problems solved using this method include the axial translation of a small particle towards a rigid square plate, an investigation of the flow field development for an axially translating small body moving through a hollow and constricted circular cylinder of finite length, the broadside motion of a circular disk through a circular cylinder and an investigation of a curious phenomenon, known as 'overshooting', in the developing axial velocity profile within a finite-length circular cylinder, given certain inlet and outlet velocity profiles.