To address the application requirements of wideband, high polarization conversion ratio (PCR) and miniaturized polarization converters, a dual-wideband and high PCR transmissive polarization converter based on chiral metasurface is designed in this paper. The proposed polarization converter consists of a three-layer structure:open rings at the top and bottom layers and a dielectric layer. The simulation results show that the metasurface uniformly achieves linear polarization conversion in the frequency ranges of 6.21—7.05 GHz and 8.74—9.73 GHz with the PCR value all exceeding 90%, and the corresponding relative bandwidths are 12.70% and 10.72%, respectively. The mechanism of high PCR and dual-band polarization conversion is revealed by analyzing the surface currents at the resonant frequencies. In addition, the influence of geometric parameters, thickness of the dielectric layer, polarization mode and polarization angle of the incident wave on the polarization manipulated characteristics of the dual-band are also studied. The proposed chiral dual-band polarization conversion metasurface possesses the advantages of simple structure and polarization insensitivity, which has potential application values in microwave communication, microscopic imaging and stealth technology.