A class of permutation tests based on appropriateU-statistics is proposed for the multisample location, scale, and association problems. TheseU-statistics need not be functions solely of the ranks of the different sample observations and for the application of the tests the parent distributions need not be continuous. The proposed tests are based on the existence of some equally likely permutations of the sample observations and are valid for small as well as large sample sizes. Further, the underlying permutation principle yields optimum estimators of the variances and covariances ofU-statistics entering into the expressions for the test statistics. The large sample test procedure is thereby simplified. This paper consists partly of expository material and partly of extensions of some allied two-sample results, considered earlier by the author (Sen [28]).