This paper is concerned with the distributions of some test statistics for a multivariate linear hypothesis under nonnormality. The test statistics considered include the likelihood ratio statistic, the Lawley-Hotelling trace criterion and the BartlettNanda-Pillai trace criterion, under normality. We derive asymptotic expansions of the null distributions of these test statistics up to the order n , where n is the sample size, under nonnormality. It is shown that our general results can be e¤ectively obtained by deriving an asymptotic expansion of the distribution of a multivariate t-statistic. As special cases of our general results our asymptotic expansions are given for Hotelling’s T 2 statistic, one-way MANOVA test statistics, etc. Numerical accuracies of asymptotic expansion approximations are examined. The validity of the expansions is also discussed. Moreover, we will find conditions such that the Bartlett correction in the normal case implies an improved w-approximation, even under nonnormality.