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This paper provides an introduction to the mathematical notion of\emph{quantum curves}. We start with a concrete example arising from a graphenumeration problem. We then develop a theory of quantum curves associated withHitchin spectral curves. A conjecture of Gaiotto, which predicts a newconstruction of opers from a Hitchin spectral curve, is explained. We give astep-by-step detailed description of the proof of the conjecture for the caseof rank $2$ Higgs bundles. Finally, we identify the two concepts of\textit{quantum curve} arising from the topological recursion formalism withthe limit oper of Gaiotto's conjecture.

A journey from the Hitchin section to the oper moduli