The global existence problem in general relativity

We survey some known facts and open questions concerning the global properties of 3+1 dimensional space--times containing a compact Cauchy surface. We consider space--times with an $\ell$--dimensional Lie algebra of space--like Killing fields. For each $\ell \leq 3$, we give some basic results and conjectures on global existence and cosmic censorship. For the case of the 3+1 dimensional Einstein equations without symmetries, a new small data global existence result is announced.