It has been shown that the cosmic censorship conjecture holds for polarized Gowdy spacetimes. In the more general, unpolarized case, however, the question remains open. It is known that cylindrically symmetric dust can collapse to form a naked singularity. Since Gowdy universes comprise gravitational waves that are locally cylindrically symmetric, perhaps these waves can collapse onto a symmetry axis and create a naked singularity. It is known that in the case of cylindrical symmetry, event horizons will not form under gravitational collapse, so the formation of a singularity on the symmetry axis would be a violation of the cosmic censorship conjecture.To search for cosmic censorship violation in Gowdy spacetimes, we must have a better understanding of their singularities. It is known that far from the symmetry axes, the spacetimes are asymptotically velocity term dominated, but this property is not known to hold near the axes. In this thesis, we take the first steps toward understanding on and near axis behavior of Gowdy spacetimes with space-sections that have the topology of the three-sphere. Null geodesic behavior on the symmetry axes is studied, and it is found that in some cases, a photon will wrap around the universe infinitely many times on its way back toward the initial singularity.