Plasmonic Fano resonance arises from the destructive interference between a superradiant and a subradiant plasmon mode that overlap spectrally with each other. Because of its importance in revealing many physical phenomena and its applications in sensing, metamaterials, photoswitching and spectroscopy, a variety of metal nanostructures have been fabricated to generate Fano resonance. However, few metal nanostructures can support deep Fano resonance with strong polarization dependence. Herein, we report on the observation of deep Fano resonance with strong polarization dependence in Au nanoplate-nanosphere heterodimers. Experiments and simulations reveal that the presence of a nanosphere at one side edge or one vertex of the nanoplate causes distinct Fano resonance. With increasing nanosphere sizes, the shape of the scattering spectrum becomes more asymmetric, with the Fano dip getting deeper correspondingly. When the nanosphere diameter reaches 68 nm, the Fano dip almost reaches the spectral background. Moreover, the heterodimers with the nanosphere attached to one vertex of the nanoplate exhibit Fano resonance with strong polarization dependence. Such heterodimers are very attractive for constructing polarization-controlled plasmonic Fano switches.