If the model does not satisfy ANOVA assumptions (normality in particular), if one-way, Kruskal-Wallis non-parametric test is recommended. But, what if you have multiple factors?
You can use a permutation test.
Form your hypothesis as a full and reduced model test and using the original data compute the F-statistic for the full and reduced model test (or another stat of interest).
Now compute the fitted values and residuals for the reduced model, then randomly permute the residuals and add them back to the fitted values, now do the full and reduced test on the permuted dataset and save the F-statistic (or other). Repeate this many times (like 1999).
The p-value is then the proportion of the statistics that are greater than or equal to the original statistic.
This can be used to test interactions or groups of terms including interactions.
The Kruskal-Wallis test is a special case of the proportional odds model. You can use the proportional odds model to model multiple factors, adjust for covariates, etc.
Friedman's test provides a non-parametric equivalent to a one-way ANOVA with a blocking factor, but can't do anything more complex than this.