Laughlin states are N-particle wave functions, successfully describing the fractional quantum Hall effect (QHE) for plateaux with simple fractions. It was understood early on, that much can be learned about QHE when Laughlin states are considered on a Riemann surface. Mathematically, it is interesting to know how do the Laughlin states depend on the Riemannian metric, magnetic potential function, complex structure moduli, singularities -- for the large number of particles N. I will review the results, conjectures and further questions in this area, and relation to topics such as Coulomb gases/beta-ensembles, Bergman kernels for holomorphic line bundles, Quillen metric, zeta determinants.