Observations of radio echoes from the moon at transit have been made from September 1953 to August 1955, using a pulsed transmitter operating at 120 Mc/s. The aerial, which is a high-gain array with a physical area of 40 square wavelengths, transmits and receives waves which are horizontally polarized. The first part of the paper discusses the theory of echoes from the moon, and includes estimates of the echo intensity and of the power spectrum of the echoes for several assumptions about the scattering function of the lunar surface. It is also shown that the ionosphere should rotate the plane of polarization of the radio waves, thus producing long-period fading of the echoes, and that the angle of rotation is a measure of the total electron content of the ionosphere. The second part of the paper presents the results of the observations. The observed echo intensity, in conjunction with the observed power spectra of the echoes, is consistent with the assumption that the moon reflects radio waves in such a way that its projected disc appears uniformly `bright', as is the case for light waves. The power reflection coefficient must be somewhat less than 0.1. The observed echoes are subject to rapid fading. This is found to be due to the moon's libration. The libration varies with the position of the moon in its orbit, causing a corresponding change in the rate of fading. Experimental proof is obtained that the slow fading of moon echoes previously reported by other workers is due to the rotation of the plane of polarization of the waves by the ionosphere. By observing the slow fading on two frequencies with a separation of 1.5 Mc/s, it is possible to measure the total angle by which the plane of polarization is rotated and hence the total electron content of the ionosphere. Some preliminary results appear to indicate that the parabolic half-thickness of the F2 region may be a more reliable measure of the extent of the ionosphere than has sometimes been supposed.