In this talk I will discuss some general aspects on quantum curves. I will emphasize the need to go beyond WKB-like perturbative schemes, and the usefulness of working with actual Hilbert spaces. I will then present three mathematical mechanisms in our theory of quantum mirror curves which make it possible to go from the discrete world of quantum mechanics to the continuous world of enumerative geometry: Fredholm determinants, large N limits, and Wigner distributions. I will also list some interesting open problems for the future.