The authors construct bilinear differential operators for Siegel modular forms mapping \(M(\Gamma)_k\times M(\Gamma)_l\) to \(M (\Gamma)_{k+l+v}\) for all even nonnegative integers \(v\) and some discrete subgroup \(\Gamma\) of \(Sp(2n,R)\) with finite covolume. These operators are called Rankin-Cohen type operators because the explicit construction of such operators has been considered by R. Rankin and H. Cohen in the genus one case. Recently, Y. Choie and W. Eholzer [J. Number Theory 68, No. 2, 160-177 (1998)], generalized this result to the case of genus two. Both of these results are special cases of the construction presented this paper. The authors also discuss a vector valued generalization of the Rankin-Cohen type operators. As they remark, one may view this paper as an attempt to describe certain spaces of invariant pluriharmonic polynomials.