This paper, authored by Azzalini and Capitanio, explores the multivariate skew-normal distribution, which extends the standard normal distribution by introducing a shape parameter to account for skewness. The authors delve into the probabilistic properties of this distribution, emphasizing its statistical relevance. They discuss inferential issues and applications in multivariate statistics, using numerical examples to illustrate their points. The paper also introduces an extension that incorporates a skewing factor into an elliptical density. The introduction highlights the tendency in statistical literature towards more flexible methods to better represent data features and reduce unrealistic assumptions. It notes that while the multivariate normal distribution is mathematically tractable, it often requires transformations to achieve multivariate normality, which can be problematic. The authors then introduce the multivariate skew-normal distribution, detailing its definition, key properties, and extensions. The paper examines the behavior of linear and quadratic forms in the skew-normal distribution, including marginal distributions, linear transforms, and conditional distributions. It provides formulas for these forms and discusses conditions for independence among components. The authors also derive cumulative generating functions and indices of skewness and kurtosis, showing how these can be used to summarize the distribution's shape. In the section on statistical issues, the authors address inferential aspects, particularly in the univariate case, and discuss the challenges of maximum likelihood estimation (MLE) due to singularities in the information matrix. They propose a reparameterization to avoid these issues and demonstrate its benefits through numerical examples. The paper concludes with applications to multivariate analysis, including fitting multivariate distributions and discriminant analysis. It provides a computational scheme for fitting the skew-normal distribution and compares it favorably to the normal distribution using diagnostic plots and likelihood ratio tests. The authors also discuss the behavior of classical multivariate techniques under the skew-normal assumption, showing that the likelihood-based discriminant function can coincide with the Fisher linear discriminant function under certain conditions.This paper, authored by Azzalini and Capitanio, explores the multivariate skew-normal distribution, which extends the standard normal distribution by introducing a shape parameter to account for skewness. The authors delve into the probabilistic properties of this distribution, emphasizing its statistical relevance. They discuss inferential issues and applications in multivariate statistics, using numerical examples to illustrate their points. The paper also introduces an extension that incorporates a skewing factor into an elliptical density. The introduction highlights the tendency in statistical literature towards more flexible methods to better represent data features and reduce unrealistic assumptions. It notes that while the multivariate normal distribution is mathematically tractable, it often requires transformations to achieve multivariate normality, which can be problematic. The authors then introduce the multivariate skew-normal distribution, detailing its definition, key properties, and extensions. The paper examines the behavior of linear and quadratic forms in the skew-normal distribution, including marginal distributions, linear transforms, and conditional distributions. It provides formulas for these forms and discusses conditions for independence among components. The authors also derive cumulative generating functions and indices of skewness and kurtosis, showing how these can be used to summarize the distribution's shape. In the section on statistical issues, the authors address inferential aspects, particularly in the univariate case, and discuss the challenges of maximum likelihood estimation (MLE) due to singularities in the information matrix. They propose a reparameterization to avoid these issues and demonstrate its benefits through numerical examples. The paper concludes with applications to multivariate analysis, including fitting multivariate distributions and discriminant analysis. It provides a computational scheme for fitting the skew-normal distribution and compares it favorably to the normal distribution using diagnostic plots and likelihood ratio tests. The authors also discuss the behavior of classical multivariate techniques under the skew-normal assumption, showing that the likelihood-based discriminant function can coincide with the Fisher linear discriminant function under certain conditions.