This note discusses some aspects of the relationship between the hypothesis that long-term bond rates follow a martingale process and the hypothesis that the bond market is efficient. It begins with some mathematics of bond prices and interest rates. It then shows that, except in one special case, the hypothesis that bond rates follow a martingale and that bond markets are efficient are theoretically inconsistent. Some empirical work is then adduced that shows that neither hypothesis is supported by the data. It concludes with some brief comments on the literature relating to this subject and some suggestions for further research.