  We review briefly the existing vertex-operator-algebraic constructions of
various tensor category structures on module categories for affine Lie
algebras. We discuss the results first conjectured in the work of Moore and
Seiberg that led us to the construction of the modular tensor category
structure in the positive integral level case. Then we review the existing
constructions and results in the following three cases: (i) the level plus the
dual Coxeter number is not a nonnegative rational number, (ii) the level is a
positive integer and (iii) the level is an admissible number. We also present
several open problems.
