We study the optical fields that are produced when two Laguerre Gauss beams, each carrying an optical vortex, are superimposed collinearly. We find that the resulting beam contains new vortices. The number of vortices and their location depends on the charge of the vortices of the component beams and the relative intensity of the two beams. Our presentation focuses on the cases where the component Laguerre-Gauss beams are of order one and two.