Algorithms for carrying out maximum likelihood parallel factor analysis (MLPARAFAC) for three-way data are described. These algorithms are based on the principle of alternating least squares, but differ from conventional PARAFAC algorithms in that they incorporate measurement error information into the trilinear decomposition. This information is represented in the form of an error covariance matrix. Four algorithms are discussed for dealing with different error structures in the three-way array. The simplest of these treats measurements with non-uniform measurement noise which is uncorrelated. The most general algorithm can analyze data with any type of noise correlation structure. The other two algorithms are simplifications of the general algorithm which can be applied with greater efficiency to cases where the noise is correlated only along one mode of the three-way array. Simulation studies carried out under a variety of measurement error conditions were used for statistical validation of the maximum likelihood properties of the algorithms. The MLPARAFAC methods are also shown to produce more accurate results than PARAFAC under a variety of conditions. Copyright © 2003 John Wiley & Sons, Ltd.