The authors obtain solutions for the title problem when the cylinder is imparted a constant velocity or a constant acceleration. It is assumed that initially both the solid obstacle and the infinite expanse of liquid surrounding it axe at rest. The velocity and acceleration imparted to the cylinder are of finite magnitude, rectilinear and in a direction perpendicular to the cylinder axis. Both solutions are expressed by means of three distinct matched expansions. These analyses are valid as long as the Reynolds number, Re, is small. Early in the processes under discussion and throughout the exterior of the cylinder, both flows are unsteady Stokesian to within a small error. Later they axe represented by inner and outer expansions. The solution structure and the nature of the expansions suggest that close to the cylinder the effects of viscosity and transiency dominate the flow field. It is only in the outer fields, and some time after motion has commenced that the effect of vorticity-convection plays a significant role. For the case in which a steady rectilinear velocity is imparted to the cylinder, both the inner and outer expansions derived here approach those which were obtained previously by Proudman and Pearson for a steady flow pattern. From the leading terms in the two expansions representing the flows close to the cylinder, approximate expressions for the time-dependent drag are obtained.