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We introduce Virasoro operators for any Landau-Ginzburg pair (W, G) where Wis a non-degenerate quasi-homogeneous polynomial and G is a certain group ofdiagonal symmetries. We propose a conjecture that the total ancestor potentialof the FJRW theory of the pair (W,G) is annihilated by these Virasorooperators. We prove the conjecture in various cases, including: (1) invertiblepolynomials with the maximal group, (2) some two-variable polynomials with theminimal group, (3) certain Calabi-Yau polynomials with groups. We also discussthe connections among Virasoro constraints, mirror symmetry of Landau-Ginzburgmodels, and Landau-Ginzburg/Calabi-Yau correspondence.

Virasoro constraints in quantum singularity theories