SUMMARY The likelihood principle is criticized and the acceptability principle is enunciated. Most acceptable estimates are defined and some techniques for their calculation are developed. In the one-parameter case the most acceptable estimator is, under some conditions, median unbiased. Conditions for the strong consistency and asymptotic normality of the estimators are obtained. These are less stringent than those for the maximum likelihood estimator. An example is given where the maximum likelihood estimator is inconsistent, but the most acceptable estimator is strongly consistent. The usual definition of consistency is criticized and a new definition suggested. Any estimator whose acceptability is bounded away from zero as the sample size n tends to infinity is strongly consistent in the new sense. The maximum likelihood estimate is shown to have high acceptability under mild conditions. Hence under these conditions it is strongly consistent in the new sense.