W-algebras are certain vertex algebras associated with nilpotent elements of a simple Lie algebra. The apparence of the AGT conjecture in physics led many researchers toward to these algebraic structures. W-algebras were introduced and developed by Zamolodchikov, Fateev-Lukyanov, Feigin-Frenkel and Kac-Roan-Wakimoto, while their finite-dimensional analogues, the finite W-algebras introduced by Premet, are important in classical problems in representation theory. They generalize both Virasoro and Kac-Moody vertex algebras. Since they are not generated by Lie algebras, the formalism of vertex algebras is necessary to study them.

The lectures will start with a short introduction to vertex algebras and basic examples. Then we will explain the construction of W-algebras that are obtained from affine vertex algebras by the BRST construction. We will also discuss some applications of equivariant W-algebras that play a key role in the quantum geometric Langlands correspondence.