This paper presents exact solutions using toroidal co-ordinates to the equations of creeping fluid motion with the no-slip boundary conditions for a toroidal particle translating in a direction normal to the axis of symmetry or rotating about an axis normal to the axis of symmetry through an otherwise infinite expanse of quiescent fluid. The associated resisting force and resisting torque are computed for toroids of various geometrical ratios b/a, b being the smallest radius of the open hole and (b + 2a) being the radius to the outermost rim of the torus. These results are compared with approximate calculations based on slender-body theory and on the theory for interacting beads. The exact and approximate calculations become asymptotically equal as b/a becomes very large, but departures from the exact calculations are apparent for b/a less than 10−100 depending on the mode of motion and the method of approximation and the approximations are unreliable for b/a less than 2·0.