  The purpose of this paper is to introduce a consistent notion of universal
and reduced crossed products by actions and coactions of groups on operator
systems and operator spaces. In particular we shall put emphasis to reveal the
full power of the universal properties of the the universal crossed products.
It turns out that to make things consistent, it seems useful to perform our
constructions on some bigger categories which allow the right framework for
studying the universal properties and which are stable under the construction
of crossed products even for non-discrete groups. In the case of operator
systems, this larger category is what we call a $C^*$-operator system, i.e., a
selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra
$A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is
given by what we call $C^*$-operator bimodules. After we introduced the
respective crossed products we show that the classical Imai-Takai and Katayama
duality theorems for crossed products by group (co-)actions on $C^*$-algebras
extend one-to-one to our notion of crossed products by $C^*$-operator systems
and $C^*$-operator bimodules.
