Topological recursion (TR) was invented as a recursive method to enumerate (with a weight) surfaces of genus g and n boundaries. Such enumerative geometry problems can also often be formulated as integrable systems (Dubrovin Zang, Givental) and eventually amount to a quantization procedure. par Initially in the EO formulation, the data needed for TR was a spectral curve: a plane complex curve, or its generalizations as local plane complex curves. par In a recent reformulation, Kontsevich and Soibelman proposed to encode the data into an algebraic structure, that they called “quantum Airy structure'', well suited for the quantization side of the story, and that could possibly allow generalizations. par We shall discuss the link between the 2, and provide a concise overview of TR and its applications, from enumerative geometry to integrable systems.

IPhT and CRM