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In 1963, Greenberg proved that every finite group appears as the monodromygroup of some morphism of Riemann surfaces. In this paper, we give twoconstructive proofs of Greenberg's result. First, we utilize free groups, whichgiven with the universal property and their construction as discrete subgroupsof $\operatorname{PSL}_2(\mathbb{R})$, yield a very natural realization offinite groups as monodromy groups. We also give a proof of Greenberg's resultbased on triangle groups $\Delta(m, n, k)$. Given any finite group $G$, we makeuse of subgroups of $\Delta(m, n, k)$ in order to explicitly find a morphism$\pi$ such that $G \simeq \operatorname{Mon}(\pi)$.

Explicit Constructions of Finite Groups as Monodromy Groups