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One of the fundamental theorems of complex dynamics is Thurston’s characterization theorem of rational maps (developed in the 1980’s in the context of a panorama of results that also includes geometrization of 3-manifolds and of surface automorphisms). Already then, John Hubbard had asked for an extension from rational to transcendental maps. The first such result, for the exponential family, was developed in joint work with Hubbard and Shishikura in the 1990’s. We present an extension of this result to the class of “structurally finite” transcendental functions. This is joint work with Sergey Shemyakov.