On the global initial value problem and the issue of singularities

In the first part of the paper we discuss what is known at present about the global initial value problem for the vacuum Einstein equations with general asymptotically flat initial data. We then give precise formulations of cosmic censorship conjectures. We also point out analogies with fluid dynamics and discuss possibilities suggested by these analogies. In the second part I discuss my work on the spherically symmetric Einstein equations with a real massless scalar field as the material model. I give an outline of the approach which has led to the proof of the conjectures in this context.