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A compound droplet subject to three-dimensional oscillatory shear flow is studied using a three-phase lattice Boltzmann model. Firstly, focusing on low values of capillary number (Ca) where the compound droplet eventually reaches steady-state oscillatory condition, we study the effect of oscillatory period, viscosities of inner and outer fluids of the compound droplet, wall confinement and Ca on the droplet behavior. As the oscillatory period increases, the maximum deformation parameters gradually approach the steady-state values in the corresponding simple shear flow for both inner and outer droplets, and the compound droplet is more synchronous with applied shear. We demonstrate for the first time that due to high pressure near two tips inside the outer droplet the inner droplet may rotate counterintuitively in a direction opposite to the outer one. The compound droplet undergoes larger deformation when either …