The action of the tail of a spermatozoon is discussed from the hydrodynamical point of view. The tail is assumed to be a flexible cylinder which is distorted by waves of lateral displacement propagated along its length. The resulting stress and motion in the surrounding fluid is analyzed mathematically. Waves propagated backwards along the tail give rise to a forward motion with velocity proportional to the square of the ratio of the amplitude of the waves to their length. The rate at which energy must be supplied to maintain the waves against the reaction of the surrounding fluid is calculated. Similar calculations for the case when waves of lateral displacement are propagated as spirals show that the body is propelled at twice the speed given it by waves of the same amplitude when the motion is confined to an axial plane. An externally applied torque is necessary to prevent the reaction of the fluid due to spiral waves from causing the cylinder to rotate. This is remarkable because the cylinder itself does not rotate. A working model of a spermatozoon was made in which spiral waves could travel down a thin rubber tube without rotating it. The torque just referred to was observed and was balanced by an eccentric weight. The performance of the model while swimming freely in glycerine was compared with the calculations. The calculated speed of the model was higher than was observed, but this discrepancy could be accounted for by the fact that the model has a body containing its motive power while the calculations refer to a disembodied tail.