  The General Theory of Relativity has been an extremely successful theory,
with a well established experimental footing, at least for weak gravitational
fields. Its predictions range from the existence of black holes, gravitational
radiation to the cosmological models, predicting a primordial beginning, namely
the big-bang. All these solutions have been obtained by first considering a
plausible distribution of matter, and through the Einstein field equation, the
spacetime metric of the geometry is determined. However, one may solve the
Einstein field equation in the reverse direction, namely, one first considers
an interesting and exotic spacetime metric, then finds the matter source
responsible for the respective geometry. In this manner, it was found that some
of these solutions possess a peculiar property, namely 'exotic matter,'
involving a stress-energy tensor that violates the null energy condition. These
geometries also allow closed timelike curves, with the respective causality
violations. These solutions are primarily useful as 'gedanken-experiments' and
as a theoretician's probe of the foundations of general relativity, and include
traversable wormholes and superluminal 'warp drive' spacetimes. Thus, one may
be tempted to denote these geometries as 'exotic' solutions of the Einstein
field equation, as they violate the energy conditions and generate closed
timelike curves. In this article, in addition to extensively exploring
interesting features, in particular, the physical properties and
characteristics of these 'exotic spacetimes,' we also analyze other non-trivial
general relativistic geometries which generate closed timelike curves.
