In this paper, spherical Luneburg lens antennas have been systematically analyzed using the Eigenfunction Expansion Method (EEM), The developed technique has capability of performing a complete 3-D analysis to characterize the multi-layered dielectric spherical lens with arbitrary permittivity and permeability. This paper describes the analysis technique, and presents the results of the parametric study of Luneburg lens antennas by varying design parameters suoh as the diameter of the lens antenna (up to 80 wavelength), number of spherical shells (up to 30 shells), air-gaps between spherical shells, and dielectric loss of the material. Many representative engineering design curves including the far-field patterns, wide-angle sidelobe characterizations, antenna efficiency have been presented.