Exact equations for the propagation of electromagnetic waves in a perfectly conducting horn of arbitrary shape are given. They take the form of an infinite set of simultaneous ordinary linear differential equations, and can be interpreted as the equations of propagation of a system of coupledE‐ andH‐waves. If the coupling is neglected, we need only consider a single differential equation for eachE‐ andH‐wave, which can be solved approximately by the W.K.B. method. This approximate solution brings out the distinction between ``transmission regions'' and ``attenuation regions'' of the horn, as found by Barrow and Chu for the sectoral horn.It is shown that the error due to neglect of coupling is, in general, of the order of the square of the flare angle as far as the variation of the field along the horn is concerned, but is of the first order in the flare angle as regards the variation of the field over the cross section. The coupling cannot, however, be neglected between modes of propagation which have the same cut‐off frequency for all cross sections. The propagation characteristics of several special shapes of horn are discussed in detail.