Abstract: 

Motivated by Virasoro conjecture for Gromov-Witten invariants, we propose three conjectures for Hodge integrals with target varieties: Virasoro conjecture, lambda_g conjecture and generalized Faber intersection number conjecture. Then we discuss the recent results on these conjectures. As applications, we show how to use   these universal constraints to derive new Dubrovin-Zhang type loop equations, which give new algorithms for computing Hodge integrals.