There are two well-known classes of integrable modules over affine Lie algebras: highest weight modules and loop modules. I will describe both classes and will explain how they generalize to give weight modules with finite-dimensional weight spaces which are not necessarily integrable. I will then explain the background for our main result — classifying the irreducible weight modules with finite-dimensional weight spaces. The classification result states that every such module is either a parabolically induced module or a loop module. The talk will assume only basic knowledge of finite-dimensional Lie algebras and their representations.