Compound drops are comprised of two of more immiscible phases. They occur in many natural and technological processes and environments, e.g. the atmosphere, liquid membranes and liquid bi-layers, direct contact heat exchange and phase separation processes. Compound liquid-liquid drop can model a living eukaryotic cell. Two-phase compound drops may consist of two touching, partially or completely engulfed parts, or have more complex geometry. Most processes involving multiphase drops are accompanied by intensive heat and mass transfer and, thus, a significant influence of Marangoni flow is anticipated, i.e. fluid motion induced variation of surface tension due to surfactant or temperature gradient along the interface. In the case of completely engulfed drop, Marangoni effect induces a relative motion of the inner particle towards the center of the outer one or in the opposite direction and, thus, stabilizes or destabilizes the liquid shell. In this chapter we overview the theoretical advances concerning the motion of multiphase compound drops of various types induced by externally imposed temperature and concentrations gradient and spontaneous Marangoni motion of such systems due to heat and mass transfer between the phases including the results of our ongoing research. Most of the works on the subject are devoted to quasi-stationary Stokes flow and non-deformable spherical interfaces corresponding to the case of small Reynolds and capillary numbers. Solutions of the Stokes equations are obtained making use of special coordinate systems (bi-spherical, toroidal, or tangent spheres), depending on the compound drop type.