A cylindrical tank, full of fluid, is oscillating with frequency ω and rotating with angular velocity Ω about its axis of symmetry. It is assumed that the amplitude of oscillation, δ, is small and the viscosity is low such that boundary layers exist. Analysis shows that the unsteady boundary layer is of thickness [ε/(1 − 2Ω/ω)]½ on the top and bottom plates and of thickness ε½ on the side walls, where ε = ν/2ω. The interior unsteady flow shows source-like behaviour at the corners. The steady flow field is caused by the steady component of the non-linear centrifugal forces coupled with an induced steady rotation of the interior. This rotation, of order δ2ω, is prograde when Ω/ω < 0·118 and retrograde otherwise. Maximum retrograde rotation occurs at Ω/ω = 0·5. A steady boundary layer of thickness [ε/(1 − 2Ω/ω)]½ exists on the top and bottom plates, and of thicknesses \[ \epsilon^{\frac{1}{2}},\quad (\nu/L^2\Omega)^{\frac{1}{3}},\quad (\nu/L^2\Omega)^{\frac{1}{4}} \] on the side walls. Experimental measurements of the interior induced steady rotation compare well with theory.