The Fuchsian--Koebe model of complex curves has far--reaching applications in geometry and number theory, perhaps most famously in the Modularity Theorem. In this talk we will review the classical uniformization theory of complex hyperbolic curves, which says that any such curve can be described as the quotient of the upper--half plane $\hyp^2$ by the action of some Fuchsian group $\Gamma < \mathrm{PSL}_2(\R)$. We will also introduce a generalization of the uniformization Theorem for real hyperbolic 3--manifolds and, if time permits, for certain p--adic curves.