  The quantum mechanics of a system of charged particles interacting with a
magnetic field on Riemann surfaces is studied. We explicitly construct the wave
functions of ground states in the case of a metric proportional to the Chern
form of the $\theta$-bundle, and the wave functions of the Landau levels in the
case of the the Poincar{\' e} metric. The degeneracy of the the Landau levels
is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin
wave function on Riemann surfaces and discuss the mathematical structure hidden
in the Laughlin wave function. Moreover the degeneracy of the Laughlin states
is also discussed.
