  In this survey, we review some of the recent connections between the
representation theory of (untwisted) quantum affine algebras and the
representation theory of current algebras. We mainly focus on the
finite-dimensional representations of these algebras. This connection arises
via the notion of the graded and classical limit of finite-dimensional
representations of quantum affine algebras. We explain how this study has led
to interesting connections with Macdonald polynomials and discuss a BGG-type
reciprocity result. We also discuss the role of Demazure modules in this theory
and several recent results on the presentation, structure, and combinatorics of
Demazure modules.
