The structured total least squares (STLS) problem is an extension of the total least squares (TLS) problem for solving an overdetermined system of equations Ax approximate to b. In many cases the extended data matrix [A b] has a special structure (Hankel, Toeplitz,...). In those cases the use of STLS is often required if a maximum likelihood (ML) estimate of x is desired. The main objective of this manuscript is to clarify the difference between several IQML-like algorithms-for solving STLS problems-that have been proposed by different authors and within different frameworks. Some of these algorithms yield suboptimal solutions while others yield optimal solutions. An important result is that the classicial IQML algorithm yields suboptimal solutions to the STLS problem. We illustrate this on a specific STLS problem, namely the estimation of the parameters of superimposed exponentially damped cosines in noise. We also indicate when this suboptimality starts playing an important role. (C) 2001 Elsevier Science B.V. All rights reserved.