Maximum likelihood techniques as applied to classification and clustering problems are examined, and the classification maximum likelihood technique, in which individual observations are assigned on an all-or-nothing basis to 1 of several classes as part of the maximization process, is shown to give results asymptotically biased. This extends Marriott''s work for normal component distributions. Numerical examples are presented for normal component distributions and for a problem in genetics. Biases apparently can be severe, though determining in simple form when the biases will and will not be severe seems difficult.