It is shown that the only Killing vector fields admitted by the Kerr-Newman spacetime are those corresponding to the time independence and axial symmetries, and that the only conformal vector fields or projective collineations admitted by the Kerr-Newman spacetime are the Killing vector fields. In addition, it is shown that the Kerr-Newman spacetime admits no covariantly constant vector fields or recurrent vector fields, and is of holonomy type R15. It is also established that any Weyl conformal collineation, Weyl projective collineation or curvature collineation admitted by the Kerr-Newman spacetime are the Killing vector fields, and that the only Ricci or matter collineations admitted by the Kerr-Newman spacetime are again Killing vector fields. Finally, some analogous results are established for the Reissner-Nordström spacetime.