Virasoro constraints for Gromov-Witten invariants have a rich history tied to the very beginning of the subject, but recently there have been many developments on the sheaf side. In this talk I will survey those developments and talk about joint work with A. Bojko and W. Lim where we propose a general conjecture of Virasoro constraints for moduli spaces of sheaves and formulate it using the vertex algebra that D. Joyce recently introduced to study wall-crossing. Using Joyce's framework we can show compatibility between wall-crossing and the constraints, which we then use to prove that they hold for moduli of stable sheaves on curves and surfaces with $h^{0,1}=h^{0,2}=0$. In the talk I will give a rough overview of the vertex algebra story and focus on the ideas behind the proof in the case of curves.