When running an ANOVA we are told that certain assumptions of the test must be present for it to be applicable to the data. I never understood the reason as to why the following assumptions were necessary for the test to function:

a) The variance of your dependent variable (residuals) should be equal in each cell of the design

b) Your dependent variable (residuals) should be approximately normally distributed for each cell of the design

I understand that there is a bit of a grey area as to if these assumptions need to be met, but for the sake of argument, if these assumptions were utterly not met in a given data set, what would be the issue with using an ANOVA?