Abstract:This talk is based on a joint paper with Prof. Kaiming Zhao. In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible modules on which each positive root vector acts locally finitely. In the second part of the paper, we obtain a necessary and sufficient condition for the tensor product of irreducible loop modules and irreducible integrable highest weight modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.