The two-dimensional theory of lunate-tail propulsion is extended to motions of arbitrary amplitude, regular or irregular, so that an accurate comparison may be made with the actual lunate-tail propulsion of scombroid fishes and cetacean mammals. There is no restriction at all on the amplitude of motion but the tail's angle of attack relative to its instantaneous path through the water is assumed to remain small. The theory is applied to the regular finite amplitude motion of a thin aerofoil with a rounded leading edge to take advantage of the suction force and a sharp trailing edge to ensure smooth tangential flow past the rear tip. This can represent the vertical motions of the horizontal lunate tails of large aspect ratio with which cetacean mammals propel themselves or the horizontal undulations of the vertical lunate tails of certain fast fishes. The dependence of the thrust, the hydromechanical propulsive efficiency and the energy wasted in churning up the eddying wake on the reduced frequency, the angle of attack and the amplitude of motion is exhibited.