The unsteady flow of rotating fluids can exhibit some fascinating and elegant phenomena. The movement of fluid due to the rotation of an encompassing container, or the flow of fluid due to the rotation of a solid object immersed within a body of fluid, is of particular interest. Specifically, we focus on the dynamics resulting from an impulsive change in the rotation rate for both these internal and external flows, including the formation of boundary layers and their subsequent evolution, and the formation of instabilities. Using a combination of analytical, computational and experimental methods, we consider the flow induced by both a torus and a sphere, contained in an otherwise quiescent body of fluid, suddenly imparted with angular momentum. Our results agree well with previous work on this classical problem, known to exhibit a boundary-layer collision process, and we are able to take this problem further to consider the post-collision dynamics, encountering some new and exciting results, such as the development and propagation of a toroidal vortex pair, as a result of the boundary-layer collision, and instability of the radial jet. We also consider the flow within fluid-filled annular and toroidal containers, when there is a sudden change in the rotation rate of the container. Using both analytical and computational approaches, we explore the initial development of the impulsively generated axisymmetric boundary layer, its subsequent instability, and the larger scale transient features within this class of flows. This allows us to compare with and, perhaps, proffer an explanation for previously unexplained experimental results for the flow within a fluid-filled torus. In particular, we demonstrate that small imperfections in the torus surface, introduced during manufacture, can generate substantial secondary motion, considerably different from that which would arise if caused by centrifugal instabilities. This serves to highlight the often overlooked impact of small disturbances on the global dynamics of such flows.