For very rapidly rotating nuclei, the moment of inertia can be readily increased if the mass is distributed in the form of a torus. Thus, a rotating toroid can be a figure of equilibrium when the angular momentum exceeds a certain limit. The angular momentum thresholds for the onset of toroidal figures of equilibrium are evaluated and found to be greater than the limits at which the fission barriers vanish for nuclei with the topology of a sphere. Stability against sausage deformation, in which the toroid becomes thicker in some sections but thinner in some other, is examined. We obtain two different results on the question of stability, depending on the flow pattern in the presence of these axially asymmetric deformations which, in turn, is greatly affected by the dissipative mechanism.[NUCLEAR STRUCTURE Rotating toroidal nuclei. Nuclei with very large angular momentum. Sausage stability. Regions of stable rotating toroidal nuclei.]