  We propose two families of tests for the classical goodness-of-fit problem to
univariate normality. The new procedures are based on $L^2$-distances of the
empirical zero-bias transformation to the normal distribution or the empirical
distribution of the data, respectively. Weak convergence results are derived
under the null hypothesis, under fixed alternatives as well as under contiguous
alternatives. Empirical critical values are provided and a comparative
finite-sample power study shows the competitiveness to classical procedures.
