This paper examines maximum likelihood techniques as applied to classification and clustering problems, and shows that the classification maximum likelihood technique, in which individual observations are assigned on an "all-or-nothing" basis to one of several classes as part of the maximization process, gives results which are asymptotically biased. This extends Marriott'ls (1975) work for normal component distributions. Numerical examples are presented for normal component distributions and for a problem in genetics. The results indicate that biases can be severe, though determining in simple form when the biases will and will not be severe seems difficult. © 1978 Biometrika Trust.