Let \(\Phi\) be a Fuchsian group on the unit disk \(\mathbb{D}\). A non-constant, meromorphic function \(f\) is called \(\Phi\)-polymorphic if, for every \(\varphi \in \Phi,\) there is a Moebius transformation \(\gamma\) such that \(f\circ\varphi = \gamma \circ f\). Defining \(f^{*}(\varphi) = \gamma,\) one obtains a homomorphism \[ f^{*}:\Phi \mapsto \text{PSL}(2,\mathbb{C}). \] The image group need not be discrete, though the authors assume it is non-elementary. The main purpose of the paper is to study the function-theoretic properties of \(f\) and its relation to the limit sets and fixed points of \(\Phi\) and its image group.