In this talk we review some existence theorems concerning the vortex solutions for some nonlinear PDE systems arising in a dually gauged harmonic map model. For the equations over a compact surface, we obtain necessary and sufficient conditions for the existence of solutions. For the equations over the full plane, we obtain all finite-energy solutions. In addition, we also present precise expressions giving the values of various physical quantities of the solutions, including magnetic charges and energies, in terms of the total numbers of vortices and antivortices, of two species and the coupling parameters involved.