This paper is concerned with optimum tests of certain composite hypotheses. In section 2 various aspects of a theorem of Scheffe concerning type $B_1$ tests are discussed. It is pointed out that the theorem can be extended to cover uniformly most powerful tests against a one-sided set of alternatives. It is also shown that the method for determining explicitly the optimum test region may in certain cases be reduced to a simple formal procedure. These results are used in section 3 to obtain optimum tests for the composite hypothesis specifying the value of the circular serial correlation coefficient in a normal distribution. A surprising feature of this example is the fact that for the simple hypothesis obtained by specifying values for the nuisance parameters no test with the corresponding optimum properties exists. In section 4 the totality of similar regions is obtained for a large class of probability laws which admit a sufficient statistic. Some composite hypotheses concerning exponential and rectangular distributions are treated in section 5. It is proved that the likelihood ratio tests of these hypotheses have various optimum properties.