  We propose a new, conjectural recursion solution for Hurwitz numbers at all
genera. This conjecture is based on recent progress in solving type B
topological string theory on the mirrors of toric Calabi-Yau manifolds, which
we briefly review to provide some background for our conjecture. We show in
particular how this B-model solution, combined with mirror symmetry for the
one-leg, framed topological vertex, leads to a recursion relation for Hodge
integrals with three Hodge class insertions. Our conjecture in Hurwitz theory
follows from this recursion for the framed vertex in the limit of infinite
framing.
