Classical families of orthogonal polynomials appear to be at the same time eigenvectors for some second order differential operators. This is the case of Hermite, Laguerre and Jacobi polynomials  on the real line. Indeed, they are the unique such examples in dimension one.  We shall discuss some higher dimensional examples. It turns out that the sets on which such may happen have a special algebraic structure, and leads to connections with Lie groups, Invariant theory, partial differential equations and probability theory.