  This article gives a synopsis on new developments in affine invariant tests
for multivariate normality in an i.i.d.-setting, with special emphasis on
asymptotic properties of several classes of weighted $L^2$-statistics. Since
weighted $L^2$-statistics typically have limit normal distributions under fixed
alternatives to normality, they open ground for a neighborhood of model
validation for normality. The paper also reviews several other invariant tests
for this problem, notably the energy test, and it presents the results of a
large-scale simulation study. All tests under study are implemented in the
accompanying R-package mnt.
