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