  After a brief review of recent rigorous results concerning the representation
theory of rational chiral conformal field theories (RCQFTs) we focus on pairs
(A,F) of conformal field theories, where F has a finite group G of global
symmetries and A is the fixpoint theory. The comparison of the representation
categories of A and F is strongly intertwined with various issues related to
braided tensor categories. We explain that, given the representation category
of A, the representation category of F can be computed (up to equivalence) by a
purely categorical construction. The latter is of considerable independent
interest since it amounts to a Galois theory for braided tensor categories. We
emphasize the characterization of modular categories as braided tensor
categories with trivial center and we state a double commutant theorem for
subcategories of modular categories. The latter implies that a modular category
M which has a replete full modular subcategory M_1 is equivalent to M_1 x M_2
where M_2=M\cap M_1' is another modular subcategory. On the other hand, the
representation category of A is not determined completely by that of F and we
identify the needed additional data in terms of soliton representations. We
comment on `holomorphic orbifold' theories, i.e. the case where F has trivial
representation theory, and close with some open problems. We point out that our
approach permits the proof of many conjectures and heuristic results on `simple
current extensions' and `holomorphic orbifold models' in the physics literature
on conformal field theory.
