Subject to certain restrictions, a characterization of unbiased estimates with minimum variance is obtained. For two fairly broad classes of problems, solutions are given which are more readily applicable. These are used to obtain such estimates in some particular cases. The applicability of the results to problems of sequential estimation is pointed out. The problem of unbiased estimation is not at present of much practical importance, but is of some theoretical interest and has been treated by many statisticians. Also, the method used in this paper may be applicable to other problems in statistics.