  The ELSV formula, first proved by Ekedahl, Lando, Shapiro, and Vainshtein,
relates Hurwitz numbers to Hodge integrals. Graber and Vakil gave another proof
of the ELSV formula by virtual localization on moduli spaces of stable maps to
the projective line, and also explained how to simplify their proof using
moduli spaces of relative stable maps to the projective line relative to a
point. In this expository article, we explain what the ELSV formula is and how
to prove it by virtual localization on moduli spaces of relative stable maps,
following Graber-Vakil. This note is based on lectures given by the author at
Summer School on "Geometry of Teichmuller Spaces and Moduli Spaces of Curves"
at Center of Mathematical Sciences, Zhejiang University, July 14--20, 2008.
