The moduli space $$\mathcal M _{g}$$ , of compact Riemann surfaces of genus $$g$$ has orbifold structure since $$\mathcal M _{g}$$ is the quotient space of the Tiechmüller space by the action of the mapping class group. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we find the orbifold structure of the moduli spaces of Riemann surfaces of genera 4 and 5.