The purpose of this paper is to try to show that certain moment conditions are essential for the admissibility of various "good" translation invariant statistical procedures. In 1951 Blackwell [1] first gave an example in which he proved that a best invariant estimate may be inadmissible. Since then many papers dealing with the admissibility of the best invariant procedures have been published. The problem of admissibility in the case of point estimation of a location parameter was treated by Blyth [2], Blackwell [1], Stein [9], [10], [11], Fox and Rubin [6], Farrell [4], Brown [3]. The problem of admissibility of certain confidence intervals was treated by Joshi [7]. The problem of admissibility in the case of a best invariant test involving a location parameter was treated by Lehmann and Stein [8]. In each paper cited above, admissibility requires the existence of one more moment than what is needed for finite risk. The first two examples of this paper indicate that, without this extra moment, inadmissibility may result. In Section 3, we show, by example, a unique best translation invariant estimate may be inadmissible if a certain moment condition fails to be satisfied. In Section 4 we prove a theorem which gives a set of sufficient conditions for the admissibility of certain translation invariant confidence interval procedures and we also give an example which shows that a certain translation invariant confidence interval procedure may be inadmissible if a certain moment condition fails to hold. In Section 5 we show by example a best translation invariant test may be inadmissible if the test is non-unique.