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Owing to the problems caused by propellers, research has turned to the biological world for inspiration for non-propeller propulsion. Rays were chosen for further study and it was found that a key feature of their swimming is the asymmetric-in-time movements of their pectoral fins. The main goal was to determine whether asymmetric-in-time oscillations produced a larger resultant force. Two flexible fins were used (NACA and biomimetic stiffness profile “BIO”). Asymmetry was defined by the proportion of the time period taken to effect one half stroke. The experiments showed that at low frequencies, asymmetric oscillation produced greater resultant force and that this force was at an angle to the chord of the fin at rest. At high frequencies, the BIO fin produced lower resultant force when oscillating asymmetrically and the angle of the resultant force was the same as for the symmetric oscillations. There was no difference between the resultant force magnitude or direction produced by the NACA fin at high frequencies. More power was used when oscillating asymmetrically but the force efficiency, the resultant force per watt, was often the same for symmetric and asymmetric oscillations. The trailing edge kinematics of the fins were analysed. Some of the kinematics variables correlated with the resultant force magnitude independently of fin type. The wake structures behind the fins oscillating at two different frequencies were examined. The wakes were geometrically asymmetric behind both fins oscillating asymmetrically at low frequency. At the higher frequency, the wakes behind the asymmetrically oscillating fins were no different to their symmetric counterparts. Asymmetric-in-time oscillation is suggested as a method of effecting turning manoeuvres and forward propulsion with only one motor. Despite the increased energetic cost, asymmetric-in-time oscillations can produce larger steady-state forces than symmetric oscillations do.