We derive some Hodge integral identities by taking various limits of the Mariño-Vafa formula using the cut-and-join equation. These identities include the formula of general
λ
g
\lambda _g
-integrals, the formula of
λ
g
−
1
\lambda _{g-1}
-integrals on
M
¯
g
,
1
{\overline {\mathcal {M}}}_{g,1}
, the formula of cubic
λ
\lambda
integrals on
M
¯
g
{\overline {\mathcal {M}}}_g
, and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the
λ
g
\lambda _g
conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Mariño-Vafa formula.