The paper describes the use of the maximum-entropy method to estimate the Doppler shift of a moving target in the presence of additive white noise or coloured noise consisting of clutter plus white noise. Computer simulation results are included which show that in a background of additive white noise a Doppler processor based on the maximum-entropy method is only slightly suboptimal with respect to a conventional Doppler processor based on the discrete Fourier transform, whereas in the presence of additive clutter with a narrow spectral width (e.g. ground clutter) it is markedly superior in performance to the conventional processor for the case of low Doppler targets.