Compound droplets raise great interests due to their applications in the pharmaceutical, cosmetic, and food industry. In spite of the growing demand of theoretical investigation of dynamics of compound droplets from those applications, very limited effort has been contributed in the analytical and/or numerical study of them. In this work, a 3D spectral boundary element method is employed to investigate the dynamics of compound droplets for both concentric and eccentric configurations. A comprehensive investigation has been carried out on the influences of the relative droplet size, relative surface tensions on the two interfaces, relative viscosities of the fluids, and the initial location of the inner droplet, on the deformation, migration, and stability of compound droplets. Two mechanisms of droplet breakup have been observed: (a) the contact of the outer and inner interface and (b) the instability of the inner droplet.