We propose a chiral metasurface (CMS) that exhibits asymmetric transmission (AT) of double circularly and linearly polarized waves at the same frequency band. In order to realize the manipulation of electromagnetic (EM) waves in the whole space, the unit cell of CMS consists of three layers of dielectric substrate and four layers of metal patches. The Z-shaped chiral micro-structure and a grating-like micro-structure are proposed and designed to achieve AT. The simulated results show that the x-polarized wave that is incident along one direction can be transmitted into the right-hand circularly polarized (RHCP) wave and the left-hand circularly polarized (LHCP) wave that is incident along the opposite direction can be reflected as the LHCP wave in the frequency band of 4.69GHz-5.84 GHz. The maximum chirality response can be reflected by AT and circular dichroism (CD) and they can reach up to 0.38 and 0.75, respectively. In addition, we also produced the sample of CMS, and the experimental results are in good agreement with the simulated results.