In factorial designs there are cases for which the number of treatments are exremely large, where the number of replication is limited.In thi dessertation at first we consider the general model with m fixed factors whose error terms are not necessarily normaly distributed and asymptotic normality is proven for ANOVA F test statistics for testing arbitary main factor effects and interactio So that whenever the number of treatments is very large then the statistics F an be used for non-normal cases as well. In the case of limited number of factor levels and large number of replications, it is well known that the F statistics normalized with respect to the degree of freedom, has asymptotically a distribution. The normalized common F-statistics for testing an effect of factor is the ratio of two quadratic forms. The quadratic form in the numerator can be decomposed into a sum of i.i.d random variables plus an asymptotically neglitible term. Thus it meets the condition for applying the central limit theorem. Since the denominator is acosistent estimator for variance, asymptotic normality is obtained for the whole term. Then we consider an asymptotic distribution for F-statistics based on {\it rank of observations}. Also we prove the asymptotical normality for two other statistics Lawley-Hoteling and Bartlet-Nada-Pillai statistics. Note that, there are no restrictions on the covariance structure of the multivariate observations and the methods can be applied to the data that which are measured in an ordinal scale. The variance terms has relatively complex forms and can be simplified either by formulating a stronger null hypothesis or by imposing an assumption on the eigenvalues of the covariance matrices or providing an upper and lower bounds on the asymptotic variance.Simulation results show that in the presence of outliers the nonparametric tests gives us better and more powerfull resultes
