The formation and breakup of an axisymmetric immiscible, viscous, laminar compound jet flowing vertically downward into and breaking up in another immiscible liquid is studied numerically. We use a front-tracking/finite difference method to track the unsteady motion and the breakup of the compound jet interfaces, which are governed by the incompressible Navier-Stokes equations for Newtonian fluids. We consider the formation and breakup of a three-fluid compound jet in which the inner fluid density is greater than the shell’s fluid density, and compare with the case when the inner fluid density is less than the shell’s fluid density. The effects of interfacial tensions in terms of Weber number are investigated. An increase in Weber number leads to an increase in the breakup length of the compound jet and a decrease in the size of compound drops.