Large amplitude oscillatory shear has been employed to study the nonlinear viscoelastic properties of three polymer melts. The resins studied included a DuPont high‐density polyethylene, a Union Carbide low‐density polyethylene, and a Dow polystyrene. The equipment used consisted of a small‐gap, concentric cylinder rheometer with a controlled‐speed motor unit and a rack‐pinion, oscillating drive mechanism. The torque was monitored by means of a torquemeter based on magnetic stress anisotropy in a steel tube and the strain was monitored by means of a displacement transducer. A signal proportional to the rate of strain was generated by integrating the strain with respect to time. Strain amplitudes up to 10 and frequencies between 0.5 and30 sec−1were employed. For purposes of material characterization, plots of stress versus rate‐of‐strain are employed. This is a closed stationary curve. Its “openness” is an indication of elasticity and its deviation from an elliptical shape is an indication of nonlinearity. Three material functions, obtainable from these curves, are defined and used for characterization. The first function is the amplitude ratio; the second function is related to the extent of the elastic component of the response, and the third one is a measure of nonlinearity. It was found that the extent of nonlinearity in the response is primarily a function of the strain amplitude rather than the frequency. The amplitude below which the response is practically linear depends strongly on the molecular structure. The response of the high‐density polyethylene was practically linear up to a strain amplitude of 10.