I will introduce the formalism of quantum Airy structures, which was proposed by Kontsevich and Soibelman as a kind of deformation quantization framework for the topological recursion. We will then see that certain twisted modules for W-algebras naturally give rise to quantum Airy structures. For W(gl_N) the latter are equivalent to the ones that can be obtained from spectral curves with ramification of arbitrary order using the Bouchard-Eynard topological recursion. I shall briefly relate these results to intersection theory on the moduli space. The talk is based on joint works with Andersen, Chekhov, Orantin, and Bouchard, Chidambaram, Noshchenko.