  We discuss some basic problems and conjectures in a program to construct
general orbifold conformal field theories using the representation theory of
vertex operator algebras. We first review a program to construct conformal
field theories. We also clarify some misunderstandings on vertex operator
algebras, modular functors and intertwining operator algebras. Then we discuss
some basic open problems and conjectures in mathematical orbifold conformal
field theory. Generalized twisted modules and their variants, their
constructions and some existence results are reviewed. Twisted intertwining
operators and their basic properties are also reviewed. The conjectural
properties in the basic open problems and conjectures mentioned above are then
formulated precisely and explicitly. Some thoughts of the author on further
developments of orbifold conformal field theory are also discussed.
