Univariate and Multivariate Generalized Hypergeometric distributions are formally defined, and under some conventional rules all possible cases are found and classified. 
Univariate distributions on the interval ［0, n］ or ［0, ∞) are clearly defined and classified into five types: A1 and A2 on finite intervals and B1, B2 and B3 on infinite intervals. On the other intervals, shifts or inversions of the distributions of the basic five types are essentially possible.
Bivariate Generalized Hypergeometric distributions are possible on the non-negative or the negative quadrant. Possible types are rather limited irrespective of the general setup. The discussions on the Bivariate distributions cover classification of the Multivariate distributions.
Singular Generalized Hypergeometric distributions are classified in the course of the discussions. Geneses of main Bivariate Generalized Hypergeometric distributions are summarized.