  The initial value problem is well-defined on a class of spacetimes broader
than the globally hyperbolic geometries for which existence and uniqueness
theorems are traditionally proved. Simple examples are the time-nonorientable
spacetimes whose orientable double cover is globally hyperbolic. These
spacetimes have generalized Cauchy surfaces on which smooth initial data sets
yield unique solutions. A more difficult problem is to characterize the class
of spacetimes with closed timelike curves that admit a well-posed initial value
problem. Examples of spacetimes with closed timelike curves are given for which
smooth initial data at past null infinity has been recently shown to yield
solutions. These solutions appear to be unique, and uniquesness has been proved
in particular cases. Other examples, however, show that confining closed
timelike curves to compact regions is not sufficient to guarantee uniqueness.
An approach to the characterization problem is suggested by the behavior of
congruences of null rays. Interacting fields have not yet been studied, but
particle models suggest that uniqueness (and possibly existence) is likely to
be lost as the strength of the interaction increases.
