This correspondence deals with the structure of the maximum-likelihood (ML) estimator for time delay with arbitrary signal and noise statistics. At high signal-to-noise ratios (SNR's), the ML estimation performs a nonlinear operation on the delayed difference of the two received waveshapes. The required nonlinearity depends only on the noise statistics. At low SNR', a closed-form simple expression for the ML, which depends only on the noise statistics and on the second-order statistics of the signal, is provided. With statistically independent noise processes, the estimator correlates two vectors generated by separate nonlinear operations on the two received waveshapes.