In this paper, nonparametric tests are presented for the hypothesis of no direct treatment effects, as well as for the hypothesis of no carryover effects, for balanced crossover designs. The number of treatments equals the number of periods p, where . The design consists of n replications of balanced crossover designs each formed by m Latin squares of order p. The tests are permutation tests and they are based on the n vectors of least squares estimators of the parameters of interest obtained from the n replications of the experiment. The exact as well as the asymptotic distribution of the test statistics are obtained and it is shown that the tests have, asymptotically, the same power as the F-ratio test.