Thurston’s characterization theorem for iterated rational maps is well known. It concerns rational maps (self-maps of the Riemann sphere with finite mapping degree) and with finite orbits of all critical points. We outline a research project, joint with Kostya Bogdanov, to extend the result to transcendental maps (infinite mapping degree) with infinite singular orbits. This is work in progress, the focus of the talk will be on the project and its challenges, rather than on completed work.