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We give a unified construction of the minimal representation of a finitecover $G$ of the conformal group of a (non necessarily euclidean) Jordanalgebra $V$. This representation is realized on the $L^2$-space of the minimalorbit $\mathcal{O}$ of the structure group $L$ of $V$. We construct itscorresponding $(\mathfrak{g},\mathfrak{k})$-module and show that it can beintegrated to a unitary irreducible representation of $G$ on$L^2(\mathcal{O})$.  In particular, we obtain a unified approach to the two most prominent minimalrepresentations, namely the Segal--Shale--Weil representation of themetaplectic group $\textup{Mp}(n,\mathbb{R})$ and the minimal representation of$\textup{O}(p+1,q+1)$ which was recently studied by T. Kobayashi, G. Mano andB. {\O}rsted.  In the second part we investigate special functions which give rise to$\mathfrak{k}$-finite vectors in the representation. Various properties ofthese special functions such as differential equations, recurrence relationsand integral formulas connect to the representation theory involved.  Finally, we define the conformal inversion operator$\mathcal{F}_{\mathcal{O}}$ by the action of the longest Weyl group element.$\mathcal{F}_{\mathcal{O}}$ is a unitary operator on $L^2(\mathcal{O})$ oforder 2. We show that the action of $\mathcal{F}_{\mathcal{O}}$ on radialfunctions is given by a special case of Meijer's $G$-transform.

Minimal representations of conformal groups and generalized Laguerre functions