Natural Exponential Families (NEF) belonging to the Grand-Babel class have variance functions (VF) of the form V$ sb{F}$(m) = P(m)$ Delta(m)$ + Q(m)$ sqrt{ Delta(m)},$ with P,Q and $ Delta$ polynomials with deg P $ geq$ 3, deg Q $ geq$ 2 and deg $ Delta$ $ geq$ 2. Although the members of this class have not as yet all been enumerated, several useful sub-classes have been fully described, namely the Morris class, with at most quadratic polynomial VFs: the Mora class, with cubic polynomial Vfs; the Babel class, with deg P = 0, deg Q $ leq$ 1 and deg $ Delta$ $ leq$ 2; and the Seshadri class, with deg P = degQ = deg$ Delta$ = 1. In order to motivate a uniform presentation of each member of these classes in compendium form, the basic properties of NEFs are surveyed, with special insistence on extension models such as convolution families, exponential dispersion models and affinities of NEFs. The Grand-Babel NEFs are presented with both a canonical parametrization which emphasizes the link to their basis measure and a more familiar or utilitarian parametrization. Expressions for the variance function, the cumulant transform, the mean-domain mapping, the density (when available), the Legendre transform and some asymptotics are given for each NEF, thus providing links to the theories of Likelihood and Quasi-likelihood, Generalized Linear Models, Saddlepoint approximation, Large deviations, Distributions and Asymptotic approximation. The notion of Canonical Caste Member (CCM), an easily identifiable representative of the equivalence class of all affinities of a NEF, is introduced; correspondingly, a table of variance functions for the CCMs of the currently classified Grand-Babel NEFs is provided.

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