The postcritical set P(f) of a holomorphic map f: P^1 -> P^1 is the union of the forward iterates of the critical points of f. When finite, Thurston's Rigidity Theorem provides a restriction on the structure of P(f), while Belyi's Theorem allows us to realize any finite algebraic set as P(f) for some f. In joint work with Sarah Koch and Curt McMullen, we studied the finite sets P(f) that arise as postcritical sets and the possible combinatorics of the induced map f: P(f) -> P(f).