  Gowdy's model of cosmological spacetimes is a much investigated subject in
classical and quantum gravity. Depending on spatial topology recollapsing as
well as expanding models are known. Several analytic tools were used in order
to clarify singular behaviour in this class of spacetimes. Here we investigate
the structure of a certain subclass, the polarized Gowdy models with spatial
T^3-topology, in the large. The asymptotics for general solutions of the
dynamical equation for one of the gravitational degrees of freedom plays a key
role while the asymptotic behaviour of the remaining metric function is a
result of solving the Hamiltonian constraint equation. Using both we are able
to prove (future) geodesic completeness in all spacetimes of this type.
