In this work, we study the broadband manipulation of polarization states of terahertz (THz) waves with flexible metamaterial both theoretically and experimentally. Firstly, we construct a chiral THz metamaterial with asymmetric L-shaped metal-dielectric-metal structure, generating a series of electric dipoles via its interacting with terahertz waves. By changing the geometric parameters of the structure, the time responses of the electric dipoles in the two orthogonal directions are effectively modulated. Consequently, the chiral metamaterial efficiently converts linearly polarized terahertz wave into a circularly polarized one. The radiation of the metamaterial remains almost unaffected by the changing of the incident angle, which indicates that this chiral metamaterial can be used to realize a flexible terahertz circularly-polarized wave plate. Further, we present the working principle of this flexible terahertz circularly-polarized wave plate at the bending state based on the equivalent circuit model. Moreover, we fabricate a flexible metamaterial wave plate by using polymers as the dielectric layer. When the linearly polarized light is incident on the metamaterial, the circularly polarized output can be achieved in a wide frequency range of 0.46–0.62 THz. The polarization conversion remains quite stable even if the sample is bent. This flexible terahertz metamaterial wave plate is expected to be applied to 6G communication, molecular detection, etc.