  We use algebraic methods to compute the simple Hurwitz numbers for arbitrary
source and target Riemann surfaces. For an elliptic curve target, we reproduce
the results previously obtained by string theorists. Motivated by the
Gromov-Witten potentials, we find a general generating function for the simple
Hurwitz numbers in terms of the representation theory of the symmetric group
S_n. We also find a generating function for Hodge integrals on the moduli space
M_{g,2} of Riemann surfaces with two marked points, similar to that found by
Faber and Pandharipande for the case of one marked point.
