I will review the existing constructions of various tensor category structures on module categories for affine Lie algebras. I will discuss the results that were first conjectured in the work of Moore and Seiberg and that led us to the construction of the modular tensor category structure in the positive integral level case. Then I will 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. I will also present several open problems.