An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.