Quantized complex curves play a central role in both topological string theory and Chern-Simons theory with complexified gauge group. In both cases these quantum curves yield operators that annihilate partition functions. However, in both cases, the actual quantization of these curves has only been understood indirectly (via matrix models on one hand, via recursion relations for the Jones polynomial on the other). I will discuss an intrinsic, geometric quantization scheme that should produce such quantum Riemann surfaces directly. N.B. Such quantum curves also show up in conformal field theory, as the operators that annihilate correlators with degenerate insertions.