Summary: We study the empirical size and power of some recently proposed tests for multivariate normality (MVN) and compare them with the existing proposals that performed best in previously published studies. We show that the J.P. Royston extension [Some techniques for assessing multivariate normality based on the Shapiro-Wilk \(W\). Appl. Stat. 32, 121–133 (1983)] to the S.S. Shapiro and M.B. Wilk test [Biometrika 52, 591–611 (1965; Zbl 0134.36501)] is unable to achieve the nominal significance level, and consider a subsequent extension proposed by J.P. Royston [Approximating the Shapiro-Wilk \(W\)-Test for non-normality. Stat. Comput. 2, 117–119 (1992)] to correct this problem, which earlier studies appear to have ignored. A consistent and invariant test proposed by N. Henze and B. Zirkler [Commun. Stat., Theory Methods 19, No. 10, 3595–3617 (1990; Zbl 0738.62068)] is found to have good power properties, particularly for sample sizes of 75 or more, while the approach suggested by J.P. Royston performs effectively at detecting departures from MVN for smaller sample sizes. We also compare our results to those of previous simulation studies, and discuss the challenges associated with generating multivariate data for such investigations.