Which polynomials p(x) in Q[x] attain values in the Fibonacci sequence F_n infinitely often? The philosophy of unlikely intersections suggests that there should be very few such polynomials, if any. In this lecture I will give the list of all such polynomials and all such rational functions. Moreover, I will discuss and present results for a related problem of reducibility: for which irreducible polynomials P(t, x) in Q(t)[x] are the specialized polynomials P(F_n, x) in Q[x] reducible for infinitely many n's? This lecture is based on my master's thesis under the supervision of Danny Neftin and Eyal Subag, and on joint work with Avi Berman and Reyad Abed Elrazik.