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Unbiasedness is probably the best known criterion for evaluating the performance of estimators. This note describes unbiasedness, demonstrating various failings of the criterion. It is shown that unbiased estimators might not exist, or might not be unique; an example of a unique but clearly unacceptable unbiased estimator is given. It is shown that unbiased estimators are not translation invariant. Various alternative criteria are described, and are illustrated through examples.