A survey on the Virasoro constraints in moduli spaces of sheaves and quiver representations

Virasoro constraints for Gromov-Witten invariants is a topic that has been tied together with the development of Gromov-Witten theory. In this paper we survey the recent developments in analogous Virasoro constraints in the parallel world of sheaf and quiver representations counting theories. We give a historical overview of the subject and explain the precise statements in the setting of moduli spaces of representations of a quiver. We discuss the wall-crossing vertex algebra, which is the main tool in the existing proofs.