  For two positive integers $m$ and $n$, we let ${\mathbb H}_n$ be the Siegel
upper half plane of degree $n$ and let ${\mathbb C}^{(m,n)}$ be the set of all
$m\times n$ complex matrices. In this article, we study differential operators
on the Siegel-Jacobi space ${\mathbb H}_n\times {\mathbb C}^{(m,n)}$ that are
invariant under the natural action of the Jacobi group $Sp(n,{\mathbb R}\ltimes
H_{\mathbb R}^{(n,m)}$ on ${\mathbb H}_n\times {\mathbb C}^{(m,n)}$, where
$H_{\mathbb R}^{(n,m)}$ denotes the Heisenberg group. We give some explicit
invariant differential operators. We present important problems which are
natural. We give some partial solutions for these natural problems.
