  We discuss coherent transport of Cooper pairs through a Cooper pair shuttle.
We analyze both the DC and AC Josephson effect in the two limiting cases where
the charging energy $E_C$ is either much larger or much smaller than the
Josephson coupling $E_J$. In the limit $E_J \ll E_C$ we present the detailed
behavior of the critical current as a function of the damping rates and the
dynamical phases. The AC effect in this regime is very sensitive to all
dynamical scales present in the problem. The effect of fluctuations of the
external periodic driving is discussed as well. In the opposite regime the
system can be mapped onto the quantum kicked rotator, a classically chaotic
system. We investigate the transport properties also in this regime showing
that the underlying classical chaotic dynamics emerges as an incoherent
transfer of Cooper pairs through the shuttle. For an appropriate choice of the
parameters the Cooper pair shuttle can exhibit the phenomenon of dynamical
localization. We discuss in details the properties of the localized regime as a
function of the phase difference between the superconducting electrodes and the
decoherence due to gate voltage fluctuations. Finally we point how dynamical
localization is reflected in the noise properties of the shuttle.
