||We conjecture Virasoro constraints for stable pair theories on Calabi-Yau 4-folds. This provides one more example of ubiquitous Virasoro constraints on sheaf-theoretic moduli spaces in various dimensions. As supporting evidence for the conjecture, we prove that Virasoro constraints for a Fano 3-fold are compatible with Virasoro constraints for a corresponding elliptic CY 4-fold. We also conjecture Virasoro constraints for moduli of torsion sheaves and study their relationship to stable pair Virasoro constraints. This is a joint work in progress with Younghan Bae and Miguel Moreira.