In this paper the translatory motion of a compound drop is examined in detail for low-Reynolds-number flow. The compound drop, consisting of a liquid drop or a gas bubble completely coated by another liquid, moves in a third immiscible fluid. An exact solution for the flow field is found in the limit of small capillary numbers by approximating the two interfaces to be spherical. The solution is found for the general case of eccentric configuration with motion of the inner sphere relative to the outer together with the motion of the system in the continuous phase. The results show that the viscous forces tend to move the inner-fluid sphere towards the front stagnation point of the compound drop. For equilibrium of the inner sphere with respect to the outer there must, therefore, be a body force towards the front. This can only be achieved with the necessary condition that there be a buoyant force on the inner sphere, opposite to that of the compound drop in the continuous phase. For a given set of fluids, two or four equilibrium configurations may be found. Of these only one or two, respectively, are stable. The others are unstable. For the special case of concentric configuration, the equilibrium is always metastable.