The density functional formalism has been used to study some properties (chemical potentials, ionization potentials, electron affinities, energies and stability) of medium and large size Na and Pb clusters within the spherical jellium model. Two versions of the density functional formalism have been used: one is the Kohn-Sham approach and the other is based on an approximate functional for the kinetic energy of the electronic cloud. Some effort has been devoted to clarify two particular points: one is the behavior of the properties listed above in the large cluster size limit, and the other is concerned with the limits for the stability of multiply charged negative clusters.