Take a branched covering map of the sphere over itself so that the forward orbit of each critical point is finite. Such maps are called Thurston maps. Examples include polynomials with well-chosen coefficients acting on the complex plane, as well as twists of these by mapping classes. Two basic problems are classifying Thurston maps up to equivalence and finding the equivalence class of a Thurston map that has been twisted. We will discuss ongoing joint work with Belk, Margalit, and Winarski that provides a new, combinatorial approach to the twisted polynomial problem. We will also propose several new research directions regarding Thurston maps. This is an oral comprehensive exam. All are welcome to attend.