Luneburg lenses and Maxwell fisheye lenses possess distinct properties of focusing, well beyond conventional lenses made of uniform materials. In this paper, a planar broadband bifunctional Luneburg-fisheye lens synthesized by gradient anisotropic metasurface is proposed. The proposed anisotropic metasurface is formed by non-resonant anisotropic cells, so that it can independently realize the equivalent gradient refractive indexes of Luneburg lens and Maxwell fisheye lens along orthogonal directions in a broad band, respectively. To verify the performance of the bifunctional lens, a prototype associated with a feeding log-periodic dipole antenna has been fabricated. Experimental results show that the proposed lens functions well over a wide frequency range with high efficiency and low profile, which coincides well with theoretical predictions and simulated results. It is expected that the proposed design will facilitate the applications of multifunctional metadevices in microwave and optical ranges.