In the study of conformal field theory and string theory, vertex operator algebras (VOAs) play a fundamental role as the mathematical foundation for the concept of chiral algebras.
One of the most important families of vertex algebras are affine vertex algebras and their associated $\mathcal{W}$-algebras, which are connected to various aspects of geometry and physics.
Recently, the affine $\mathcal{W}$-algebras associated to the subregular nilpotent element $\mathcal{W}^k(\mathfrak{g}, f_{sub})$ have attracted a lot of interest.
In this talk I will introduce some basic notions in the theory of vertex algebras and present some recent results about the structure and representation theory of certain subregular $\mathcal{W}$-algebras at integer levels.