  A general overview of the phenomenon of automatic continuity of homomorphisms
between Polish groups is given. In particular, we study variants and
improvements of the closed graph theorem, applying these to the problem of
continuity of universally measurable homomorphisms and also to gauge the amount
of choice needed to construct discontinuous homomorphisms between Polish
groups. Furthermore, we provide a simple proof of automatic continuity in the
context of homeomorphism groups of compact manifolds and a complete reworking
of automatic continuity theory in the context of isometry groups of highly
homogeneous complete metric structures.
