In this paper, the asymptotic and permutation testing procedures are developed in general factorial designs without assuming homoscedasticity or a particular error distribution. The one-way layout, crossed and hierarchically nested designs are contained in our general framework. New test statistics are modifications of Wald-type statistic, where a weight matrix is a certain diagonal matrix. Asymptotic properties of the new solutions are also investigated. In particular, the consistency of the tests under fixed alternatives or asymptotic validity of the permutation procedures are proved in many cases. Simulation studies show that, in the case of small sample sizes, some of the proposed methods perform comparably to or even better in certain situations than the Wald-type permutation test of Pauly et al. (2015). Illustrative real data examples of the use of the tests in practice are also given.