  An important question in mathematical relativity theory is that of the nature
of spacetime singularities. The equations of general relativity, the Einstein
equations, are essentially hyperbolic in nature and the study of spacetime
singularities is naturally related to blow-up phenomena for nonlinear
hyperbolic systems. These connections are explained and recent progress in
applying the theory of hyperbolic equations in this field is presented. A
direction which has turned out to be fruitful is that of constructing large
families of solutions of the Einstein equations with singularities of a simple
type by solving singular hyperbolic systems. Heuristic considerations indicate,
however, that the generic case will be much more complicated and require
different techniques.
