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The motion, deformation, and stability of compound droplets in extensional flows are investigated numerically via a three-dimensional spectral boundary element method. We examine the droplet stability under the influences of the capillary number, the inner droplet size and the relative magnitude of the surface tension of the two interfaces composing the compound droplet. The influence of viscosity on the droplet deformation is also discussed. We conclude that a compound droplet with a larger inner droplet and/or smaller inner surface tension is less stable and cannot withstand strong flow. For moderate viscosity ratios, a compound droplet with a more viscous 'shell' exhibits larger deformation at steady state. In addition, for an eccentric compound droplet, both the inner and outer droplets tend to migrate away from its original location due to the asymmetry of the problem. The initial location of the inner droplet also influences the droplet stability as well as the migration velocity of the compound droplet. [ABSTRACT FROM AUTHOR]