Abstract: The Lie algebra $gl(\infty)$ consisting of infinite matrices with finitely many nonzero entries is an infinite dimensional Lie algebra which is a natural generalization of (simple) finite dimensional Lie algebras. It inherits a lot of the properties of the Lie algebras $gl_n$ but also exhibit many new features. One significant difference is that there are several possible generalizations of finite dimensional representations. The goal of this talk is to motivate the study of integrable weight modules, to discuss their properties, and to provide a classifications of the irreducible integrable weight modules with finite dimensional weight