Ensembles of collectively moving particles like flocks of birds, bacteria, or filamentous polymers show a broad range of intriguing phenomena, yet seem to obey very similar physical principles. These generic principles have been predicted to lead to characteristic density fluctuations, which are in sharp contrast to normal fluctuations determining the properties of ordered systems in thermal equilibrium. Using high-density motility assays of driven filaments, we characterize here the origin and nature of giant fluctuations that emerge in this class of systems. By showing that these unique statistical properties result from the coupling between particle density and the topology of the velocity field of the particles, we provide insight in the physics of collective motion.active fluids | nonequilibrium T he composition of active systems that show collective motion is quite generic: They consist of a sufficiently high density of "particles" like birds (1), insects (2), or fish (3), vibrated granules (4, 5), bacteria and cells (6-10), or filamentous proteins (11-17) that are either self-propelled or actively driven. However, the dynamics that results from mostly local interactions among the driven constituents (18) is anything but simple: Apart from collective motion and nematic or polar order, these systems can show such intriguing phenomena as swirling motion (5,6,12,15), spontaneous and collective changes in the direction of motion (1, 3, 11), and persistent density inhomogeneities (7,11,12,16) and can generate macroscopic fluid flow (17). The theoretical description of these systems proves exceedingly difficult. To elucidate the underlying physical principles and to assess whether these systems rely on unifying organizing principles, numerous theoretical studies have been devoted to model (self-)propelled particle systems. Pioneered by the work of Vicsek et al. (19), they approach the problem on all levels of description ranging from agent-based simulations (20-24) and mesoscopic models coarsegraining microscopic interaction rules (25-29) to mean field models in the hydrodynamic limit (10,(30)(31)(32).The dynamic properties of all these models are strikingly similar to the phenomena observed in this broad class of systems. However, due to the lack of adequate and well-controlled experimental systems, a quantitative comparison with experimental results so far proved difficult. Of particular interest are certain key properties of collective motion that were first identified by Toner and Tu (30). These hallmarks of collective motility include the occurrence of abnormally large fluctuations in the particle density, the so-called giant number fluctuations (4,7,16,24,30,31,33) and correlations that are anisotropic with respect to the direction of collective motion (7,20,30,34).Despite the utmost importance of these unifying observables for a sound understanding of the physics of collective motion, most experimental systems studied to date defy their unambiguous measurement. On the one hand, this can be attributed to t...