A fundamental invariant associated to every complex rational function f is its monodromy group, that is, the Galois group of the covering $f:P^1->P^1$ on the projective line $P^1$. We shall discuss the accumulating work towards determining the degree n complex rational functions with a specified monodromy group that is not $A_n$ or $S_n$, and its applications to problems in number theory, complex analysis, and complex dynamics. Joint work with Michael Zieve.