We discuss relations between two a priori unrelated classes of objects: (i) W-algebras, which are certain symmetry algebras of two-dimensional conformal field theories, and (ii) four-dimensional N=2 gauge theories in the presence of surface operators (certain two-dimensional defects). In particular, we relate the classifications of W-algebras and surface operators.

We review the recently uncovered connections between three classes of theories: A_r quiver matrix models, d=2 conformal A_r Toda field theories and d=4 N=2 conformal A_r quiver gauge theories

After a general overview, the construction of D-brane boundary states in the pure spinor superstring will be discussed.