Many Hodge integral identities, including the ELSV formula of Hurwitz numbers and the lambda_g conjecture, are various limits of the formula of one-partition Hodge integrals conjectured by Marino and Vafa. Local Gromov-Witten invariants in all degrees and all genera of any toric surfaces in a Calabi-Yau threefold are determined by the formula of two-partition Hodge integrals. I will describe proofs of the formulae of one-partition and two-partition Hodge integrals based on joint works with Kefeng Liu and Jian Zhou.