Quantum curves are a magical object. It is conjectured that they capture the information of quantum topological invariants in an effective and compact way. The relation between quantum curves and the Eynard-Orantin theory was first exemplified in an influential paper of Gukov and Sułkowski. In the first part of this talk, for introduction, I will present the simplest and mathematically elegant example of quantum curves and the Eynard-Orantin formalism based on the Catalan numbers. (This part is based on my joint paper with P. Sułkowski and an earlier paper with Dumitrescu et al.) In the second part I will explain the construction of quantization of the spectral curves appearing in the theory of Hitchin fibrations. The main theorem is that the Eynard-Orantin theory indeed provides a mechanism of constructing the canonical generator of a D-module on an arbitrary compact Riemann surface. The semi-classical approximation of this D-module coincides with the Hitchin spectral curve. (This part is based on my joint work with O. Dumitrescu of Hanover.) The talk will be given in an elementary and pedagogical language.