  Quantum curves were introduced in the physics literature. We develop a
mathematical framework for the case associated with Hitchin spectral curves. In
this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth
projective algebraic curve, whose semi-classical limit produces the Hitchin
spectral curve of a Higgs bundle. We give a method of quantization of Hitchin
spectral curves by concretely constructing one-parameter deformation families
of opers.
  We propose a variant of the topological recursion of Eynard--Orantin and
Mirzakhani for the context of singular Hitchin spectral curves. We show that a
PDE version of topological recursion provides all-order WKB analysis for the
Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral
curves associated with meromorphic $SL(2,\mathbb{C})$-Higgs bundles.
Topological recursion can be considered as a process of quantization of Hitchin
spectral curves. We prove that these two quantizations, one via the
construction of families of opers, and the other via the PDE recursion of
topological type, agree for holomorphic and meromorphic
$SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the
Airy differential equation provides a typical example. Through these classical
examples, we see that quantum curves relate Higgs bundles, opers, a conjecture
of Gaiotto, and quantum invariants, such as Gromov--Witten invariants
