Normality of residuals or variables

[TomiSzucs]

We currently use multivariate normality testing in regression. What is the exact relationship between the multivariate normality of variables and the normality of the residuals? Should we use both, should we switch?
A handy aspect of multivariate normality: Linearity tests in regression #260

The literature is not clear on which normality is assumed in simple and multiple linear regression.