This paper deals with maximum-likelihood system identification when both the input and the output signals are corrupted by Gaussian observation noise. A derivation of exact maximum-likelihood estimation for this problem is included, but the difficulty of implementing it numerically precludes its practical evaluation at this time. A new approximate method is introduced, called the ‘output reference’ method, in which the input noise is referred to the output, and an iterative gradient search method used. This technique requires noa prioriknowledge of the noise covariance matrix. The method of Koopmans—Levin, which does require knowledge of the noise covariance matrix, is then reviewed in detail, and experimental results are presented for the white noise case which indicate that the output reference method is more accurate.