This monograph is a collective effort by Hamedani and Ghosh which focuses on a topic which has appeared in many forms: journal articles, short reviews, books; i.e., characteri- zations of bivariate, multivariate and matrix variate normal distributions. The first author is thankful to the second author for his continuous encouragement of invaluable suggestions in preparing this short monograph. This monograph attempts to collect all the characteriza- tions results related to a Gaussian distribution in bivariate and in higher domains that exist in the literature consequently claims its uniqueness. It also tries to paint some sketches of the proofs to some of the characterization results that are not available in the literature as of yet to the best of the knowledge of the authors. However, the authors do believe that contributors to this subject matter might have these ideas in their mind but did not feel nec- essary to pen it down. While this monograph heavily draws on articles by the second author, Hamedani and his co-authors in a series of papers, we sincerely hope that the readers and practitioners in need of an appropriate Gaussian distribution will find it very useful.
