A series of uniaxial tests (including multilevel loading–unloading recovery, creep-recovery, and cyclic tension–compression/tension ones) were performed to investigate the monotonic and cyclic viscoelastic–viscoplastic deformations of polycarbonate (PC) polymer at room temperature. The results show that the PC exhibits strong nonlinearity and rate-dependence, and obvious ratchetting occurs during the stress-controlled cyclic tension–compression/tension tests with nonzero mean stress, which comes from both the viscoelasticity and viscoplasticity of the PC. Based on the experimental observation, a nonlinear viscoelastic–viscoplastic cyclic constitutive model is then constructed. The viscoelastic part of the proposed model is constructed by extending the Schapery's nonlinear viscoelastic model, and the viscoplastic one is established by adopting the Ohno–Abdel-Karim's nonlinear kinematic hardening rule to describe the accumulation of irrecoverable viscoplastic strain produced during cyclic loading. Furthermore, the dependence of elastic compliance of the PC on the accumulated viscoplastic strain is considered. Finally, the capability of the proposed model is verified by comparing the predicted results with the corresponding experimental ones of the PC. It is shown that the proposed model provides reasonable predictions to the various deformation characteristics of the PC presented in the multilevel loading–unloading recovery, creep-recovery, and cyclic tension–compression/tension tests.