  In this article we introduce a theory of integration for deterministic,
operator-valued integrands with respect to cylindrical L\'evy processes in
separable Banach spaces. Here, a cylindrical L\'evy process is understood in
the classical framework of cylindrical random variables and cylindrical
measures, and thus, it can be considered as a natural generalisation of
cylindrical Wiener processes or white noises. Depending on the underlying
Banach space, we provide necessary and/or sufficient conditions for a function
to be integrable. In the last part, the developed theory is applied to define
Ornstein-Uhlenbeck processes driven by cylindrical L\'evy processes and several
examples are considered.
