The emergent dynamics of collective cellular movement are typically thought to depend on how cells interact with one another and the mechanisms used to drive motility, both of which exhibit remarkable diversity across different biological systems. Here we report experimental evidence of a universal feature in the patterns of flow that spontaneously emerge in groups of collectively moving cells. Specifically, we demonstrate that the flows generated by collectively moving dog kidney cells, human breast cancer cells and two different strains of pathogenic bacteria exhibit robust conformal invariance. We also show that the precise form of invariance in all four systems is described by the Schramm-Loewner evolution-a family of planar curves defined by a single parameter-and belongs to the percolation universality class. The presence of universal conformal invariance reveals that the macroscopic features of living biological matter exhibit universal translational, rotational and scale symmetries that are independent of the microscopic properties of its constituents. Our results show that flow patterns generated by different systems are highly conserved and that biological systems can be used to experimentally test predictions from the theories for conformally invariant structures.