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.