Singularities ~n General Relativity are due to incompleteness of space-time. This thesis examines the relationships between some of the different notions of incompleteness of a manifold with connection, together with the stability of geodesic completeness and incompleteness. Some p(parallelisation)-completions of Rand S1 are compared with the b(bundle)-completions, and the applicability of the p-boundary construction to General Relativity is discussed. Relationsips are shown between g(geodesic), b(bundle) and b.a.(bounded acceleration) incompleteness. Further acceleration is dependent notions of incompleteness are defined, and it shown that A they are all equivalent to the existing notions of completeness for a Riemannian manifold. The Whitney C K topologies provide a way of topologising the space of metrics on a manifold, in order to consider stability of geodesic completeness or incompleteness. It is shown how the spaces of connections and sprays may also be topologised, and the continuity of some important mappings is demonstrated. It turns out that for R both geodesic completeness and incompleteness are stable with respect to perturbation of the spray. Incompleteness of st ~s also stable, but the complete sprays are closed. For S1~S1 and R~S1 it is shown that null and time like geodesic completeness and incompleteness all fail to be stable with respect to the space of Lorentz metrics. Given a connection/spray on a pair of manifolds, one can construct a connection/spray on their product, and this is used to show how instability of completeness/incompleteness may arise if the product ~s compact. It is also shown how to induce sprays on the retraction of a manifold.