The theory of a parametric travelling-wave amplifier is developed, including the pumping wave explicitly.Particular reference is made to two characteristics of such amplifiers which theories assuming a constant pumping-wave amplitude fail to predict.The first arises from ohmic attenuation in the line, and limits the useful length of line to a length of the order of (α1α3)−1/2, where α1and α3are the signal- and pumping-wave attenuation coefficients. Further increase in length leads to a decrease in gain.The second characteristic is a saturation effect at high signal levels. This is found to be associated with a spatially periodic power transfer between signal and idling waves on the one hand and pumping wave on the other. The maximum power output at the signal frequency is obtainable from the input pump power and the input signal power through the Manley-Rowe relations.