In Newtonian physics and special relativity it is reasonable to demand that classical models of continuous matter exhibit the global existence property - that all solutions to the field equations evolving from sufficiently regular initial conditions should persist, uniquely for all time without developing singularities. In general relativity however the space-time itself can become singular in a finite time and well-known examples of this behavior are believed, on good evidence, to be stable and not the artifacts of special symmetry. Nevertheless there is a natural analogue in general relativity to the global existence concept, namely Penrose's cosmic censorship conjecture.
In this lecture we shall review some of the methods for establishing global existence when it holds and then turn to the question of whether such methods could perhaps be sufficiently refined so as to deal with the more subtle issue of cosmic censorship. After recalling some basic energy arguments and their higher order generalizations we focus on so-called light-cone estimates and their successful application to the proof of global existence for a variety of interesting systems including the Yang-Mills (-Higgs) equations in both flat and curved background 4-dimensional space-times. We then discuss the extent to which such techniques can be extended and applied to the basic curvature propagation equation in general relativity.