Multi-beam microwave antennas have attracted enormous attention owing to their wide range of applications in communication systems. Here, we propose a broadband metamaterial-based multi-beam Luneburg lens-antenna with low polarization sensitivity. The lens is constructed from additively manufactured spherical layers, where the effective permittivity of the constituting elements is obtained by adjusting the ratio of dielectric material to air. Flexible microstrip patch antennas operating at different frequencies are used as primary feeds illuminating the lens to validate the radiation features of the lens-antenna system. The proposed Luneburg lens-antenna achieves ±72° beam scanning angle over a broad frequency range spanning from 2 GHz to 8 GHz and presents a gain between 15.3 dBi and 22 dBi, suggesting potential applications in microwave- and millimeter-wave mobile communications, radar detection and remote sensing.