In this talk I will discuss random matrices that are matrix analogs of the well known Meixner laws: gaussian, Poisson, gamma, Pascal (negative binomial), hyperbolic secant, and binomial.

The defining property selected for the matrix extension is a characterization of the Meixner laws by the quadratic conditional variances due to Lukacs and Laha. Since only the case of $2\times 2$ real/comples/symplectic ensembles is completely resolved, there are numerous questions left to investigate.

The talk is based on a joint paper with Gérard Letac.