Extensive experimental investigations at ambient temperature on commercial Nylon 66, PEI (Polyetherimide) and PEEK (poly(ether ether ketone)) have shown that the overstress model developed for viscoplasticity should be, in principle, capable of modeling for solid polymers the rate-dependent behavior, including creep, relaxation and cyclic motions. The viscoplasticity theory based on overstress was modified accordingly to allow for the modeling of typical solid polymer deformation behavior. Included are nonlinear rate sensitivity, curved unloading, significant strain recovery at zero stress and cyclic softening. The visco plasticity theory based on overstress for polymers (VBOP) is introduced in uniaxial formulation. It is shown that VBOP can be thought of as a modified standard linear solid with overstress-dependent viscosity and nonlinear, hysteretic equilibrium stress evolution. VBOP consists of a flow law that is easily adopted to cases where the strain or the stress is the independent variable. The flow law depends on the overstress, the difference between the stress and the equilibrium stress with the latter being a state variable of VBOP. The growth law of the equilibrium stress in turn contains the kinematic stress and the isotropic or rate-independent stress, two additional state variables of VBOP. The material constants of VBOP are determined for Nylon 66 at room temperature and various tests are simulated by numerically integrating the set of nonlinear differential equations. The simulations include monotonic loading and unloading at various strain rates, repeated relaxation, recovery at zero stress that is dependent on prior strain rate, and cyclic strain-controlled loading. Finally, the stress-controlled loading and unloading are predicted with very good results. The simulations and predictions show that VBOP is competent at modeling the behavior of Nylon 66 and other solid polymers.