Measures of multivariate skewness and kurtosis are developed by extending certain studies on robustness of the t statistic. These measures are shown to possess desirable properties. The asymptotic distributions of the measures for samples from a multivariate normal population are derived and a test of multivariate normality is proposed. The effect of nonnormality on the size of the one-sample Hotelling's T2 test is studied empirically with the help of these measures, and it is found that Hotelling's T2 test is more sensitive to the measure of skewness than to the measure of kurtosis.