  We introduce and define the quantum affine $(m|n)$-superspace (or say quantum
Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann
superalgebra $\Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal
W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential
operators (QDO for short) algebra $\textrm{Diff}_q(\Omega_q)$ defined over
$\Omega_q(m|n)$, which is a smash product of the quantum differential Hopf
algebra $\mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum
Manin superspace) and the quantum Grassmann superalgebra $\Omega_q(m|n)$. An
interested point of this approach here is that even though $\mathcal
W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some
interesting sub-quotients existed inside. This point of view gives us one of
main expected results, that is, the quantum (restricted) Grassmann superalgebra
$\Omega_q$ is made into the $\mathcal U_q(\mathfrak g)$-module (super)algebra
structure,$\Omega_q=\Omega_q(m|n)$ for $q$ generic, or $\Omega_q(m|n, \bold 1)$
for $q$ root of unity, and $\mathfrak g=\mathfrak{gl}(m|n)$ or $\mathfrak
{sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach
provides us with explicit realization models for some simple $\mathcal
U_q(\mathfrak g)$-modules, together with the concrete information on their
dimensions. Similar results hold for the quantum dual Grassmann superalgebra
$\Omega_q^!$ as $\mathcal U_q(\mathfrak g)$-module algebra.In the paper some
examples of pointed Hopf algebras can arise from the QDOs, whose idea is an
expansion of the spirit noted by Manin in \cite{Ma}, \& \cite{Ma1}.
