  Considering quantum Hall states on geometric backgrounds has proved over the
past few years to be a useful tool for uncovering their less evident
properties, such as gravitational and electromagnetic responses, topological
phases and novel geometric adiabatic transport coefficients. One of the
transport coefficients, the central charge associated with the gravitational
anomaly, appears as a Chern number for the adiabatic transport on the moduli
spaces of higher genus Riemann surfaces. This calls for a better understanding
of the QH states on these backgrounds. Here we present a rigorous definition
and give a detailed account of the construction of Laughlin states on Riemann
surfaces of genus ${\rm g}>1$. By the first principles construction we prove
that the dimension of the vector space of Laughlin states is at least
$\beta^{\rm g}$ for the filling fraction $\nu=1/\beta$. Then using the path
integral for the 2d bosonic field compactified on a circle, we reproduce the
conjectured $\beta^{\rm g}$-degeneracy as the number of independent holomorphic
blocks. We also discuss the lowest Landau level, integer QH state and its
relation to the bosonization formulas on higher genus Riemann surfaces.
