  This thesis reviews minimal N=2 chiral supergravities coupled to matter in
six dimensions with emphasis on anomaly cancellation. In general,
six-dimensional chiral supergravities suffer from gravitational, gauge and
mixed anomalies which render the theories inconsistent at the quantum level.
Consistency is restored if the anomalies of the theory cancel via the
Green-Schwarz mechanism or generalizations thereof. The anomaly cancellation
conditions translate into a certain set of constraints for the gauge group of
the theory as well as on its matter content. For the case of ungauged theories
these constraints admit numerous solutions but, in the case of gauged theories,
the allowed solutions are remarkably few. In this thesis, we examine these
anomaly cancellation conditions in detail and we present all solutions to these
conditions under certain restrictions on the allowed gauge groups and
representations, imposed for practical reasons. We also briefly examine anomaly
cancellation in the context of Horava-Witten-type compactifications of minimal
seven-dimensional supergravity. Finally, we discuss some basic aspects of 4D
compactifications of the gauged models.
