Abstract: In the 90s, Kazhdan and Lusztig established an equivalence between certain affine Lie algebra representations and representations of the Lusztig quantum group. We will begin by defining these objects, and review some ingredients of their proof. After that, we will explain some ideas from our joint work with Lin Chen, where we proved an extension of this theorem using factorization algebras. Though our methods are different from the original proof, we will try to draw some connections between the two, such as the crucial role played by conformal blocks in both cases. Time permitting, we will also explain how some of these ideas play a role in the recently announced proof (by Gaitsgory et al.) of the unramified geometric Langlands conjecture.