In this thesis we examine two main problems. Firstly, we attempt to match the most general cylindrically symmetric vacuum spacetime with a Robertson-Walker interior. The matching conditions show that the interior must be dust filled, the boundary must be comoving and the vacuum region must be polarized. We use a result of Thorne's to simplify the line element. We can then prove that the matching is impossible. This demonstrates the impossibility of generalising the Oppenheimer-Snyder model of gravitational collapse to the cylindrically symmetric case The second problem is an analysis of cylindncally symmetric spacetimes with self-similarity modelling gravitational collapse The field equations and regularity conditions are examined firstly for a vacuum spacetime and then for a dust filled spacetime. The vacuum case leads to an explicit solution but no solutions that are of relevance to gravitational collapse. In the dust case, the solution of the field equations reduces to the solution of a non-linear third-order ordinary differential equation. A dynamical systems approach is then adopted, and an autonomous three-dimensional system is obtained. A unique solution is found to emanate from the regular axis {r = 0, t < 0}, where t and r are time and radial coordinates which emerge naturally from the analysis This solution persists up to {t = 0, r > 0}, which we define as Co The solution coming from Co has one parameter (a bifurcation has occurred) and propagates up to the future null cone, 3, through the scaling origin p,, where p, = {(r, t) = (0, 0)). We describe the physical invariants of the system and discuss the nature of such a spacetime in terms of its global structure.