The main focus of this work was to investigate the nature of unsteady boundary-layer development over finite domains, with the behaviour of the boundary layer on a rotating sphere in an unbounded, rotating fluid used as a prototype. The sphere and its surrounding fluid are assumed to be initially rotating as a solid body, and the evolution of a boundary layer on the sphere is analysed in cases where the sphere has been smoothly slowed, or brought to a state of rotation in an opposite sense to its initial conditions.It may be seen that a characteristic property of this flow is that the boundary layer is bi-directional; over most of the streamwise domain for the flow, whether the flow is positive or negative in the streamwise coordinate direction depends on the transverse location being considered. This fact leads to challenges in the numerical evaluation of the flow field due to the parabolic nature of the boundary-layer equations. A further consideration is the implication that these regions of reversed flow cause the flow field to contain minima and maxima in the streamwise velocity component. This has been shown in a little-known study by Cowley et al. (1985) to cause the boundary layer to become susceptible to asymptotically short-scale perturbations with large frequencies. The unsteady boundary layer on a rotating sphere under these conditions is consequently shown to be extremely challenging to compute numerically. It is also found that using local approximations at the ends of the finite domain, which in the case of the sphere are the pole and equator, to investigate the two-dimensional boundary layer can cause difficulties, as in some cases there exist steady, spatial perturbations to a boundary-layer state which introduce short spatial scales.The instabilities and other features analysed in this work are framed largely in the context of the rotating sphere, but the causes of the phenomena are found to be sufficiently generic that they may be observed in other physical contexts. To demonstrate this, the shallow katabatic flow down a cooled slope is briefly investigated, and the above mathematical features are again uncovered.