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