Vertex algebras of chiral differential operators on a complex reductive group G are "Kac-Moody" versions of the usual algebra of differential operators on G. Their categories of modules are especially interesting because they are related to the theory of D-modules on the loop group of G. That allows one to reformulate some conjectures of the (quantum) geometric Langlands program in the language of vertex algebras. For instance, in view of the geometric Satake equivalence, one may expect the appearance of the category of representations of the Langlands dual group of G. In this talk, I will define this family of vertex algebras and we will see that they are classified by a certain parameter called level. Then, for generic levels, we will see that "to find" the Langlands dual group, it is necessary to perform a quantum Hamiltonian reduction. Finally, I will build simple modules on the closely related equivariant W-algebra that match the combinatorics of the Langlands dual group.