The paper discusses the statistical applications of the multivariate skew-normal distribution, introduced by Azzalini & Dalla Valle (1996). This distribution extends the normal distribution by adding a shape parameter to regulate skewness. The paper examines probabilistic properties, inferential issues, and applications to multivariate statistics. It also describes a further extension introducing a skewing factor of an elliptical density. The multivariate skew-normal distribution is defined by a density function involving a normal density and a cumulative distribution function. Key properties include the mean vector, variance matrix, and a stochastic representation for generating random numbers. The distribution is flexible and maintains many properties of the normal distribution, such as the validity of linear and quadratic forms. The paper explores the behavior of the skew-normal distribution in linear and quadratic forms, showing that it retains some properties of the normal distribution. It discusses marginal distributions, linear transforms, and conditions for independence among components. Quadratic forms are also examined, with results showing that they follow chi-squared distributions under certain conditions. The paper also addresses cumulants and indices of skewness and kurtosis, providing expressions for these measures in the context of the skew-normal distribution. It introduces location and scale parameters, extending the distribution to account for these aspects. The paper discusses conditional distributions, showing that they can be approximated by skew-normal densities, which is useful for statistical inference. It also addresses issues in the scalar case, including the challenges of maximum likelihood estimation and the use of alternative parametrizations to avoid singularities in the information matrix. Applications to multivariate analysis are presented, including fitting multivariate distributions and discriminant analysis. The paper demonstrates the effectiveness of the skew-normal distribution in modeling real data, showing improved performance compared to the normal distribution in certain cases. The results highlight the flexibility and utility of the skew-normal distribution in statistical modeling and analysis.The paper discusses the statistical applications of the multivariate skew-normal distribution, introduced by Azzalini & Dalla Valle (1996). This distribution extends the normal distribution by adding a shape parameter to regulate skewness. The paper examines probabilistic properties, inferential issues, and applications to multivariate statistics. It also describes a further extension introducing a skewing factor of an elliptical density. The multivariate skew-normal distribution is defined by a density function involving a normal density and a cumulative distribution function. Key properties include the mean vector, variance matrix, and a stochastic representation for generating random numbers. The distribution is flexible and maintains many properties of the normal distribution, such as the validity of linear and quadratic forms. The paper explores the behavior of the skew-normal distribution in linear and quadratic forms, showing that it retains some properties of the normal distribution. It discusses marginal distributions, linear transforms, and conditions for independence among components. Quadratic forms are also examined, with results showing that they follow chi-squared distributions under certain conditions. The paper also addresses cumulants and indices of skewness and kurtosis, providing expressions for these measures in the context of the skew-normal distribution. It introduces location and scale parameters, extending the distribution to account for these aspects. The paper discusses conditional distributions, showing that they can be approximated by skew-normal densities, which is useful for statistical inference. It also addresses issues in the scalar case, including the challenges of maximum likelihood estimation and the use of alternative parametrizations to avoid singularities in the information matrix. Applications to multivariate analysis are presented, including fitting multivariate distributions and discriminant analysis. The paper demonstrates the effectiveness of the skew-normal distribution in modeling real data, showing improved performance compared to the normal distribution in certain cases. The results highlight the flexibility and utility of the skew-normal distribution in statistical modeling and analysis.