Fundamental biological processes including morphogenesis, tissue repair and tumour metastasis require collective cell motions, and to drive these motions cells exert traction forces on their surroundings. Current understanding emphasizes that these traction forces arise mainly in `leader cells' at the front edge of the advancing cell sheet. Our data are contrary to that assumption and show for the first time by direct measurement that traction forces driving collective cell migration arise predominately many cell rows behind the leading front edge and extend across enormous distances. Traction fluctuations are anomalous, moreover, exhibiting broad non-Gaussian distributions characterized by exponential tails. Taken together, these unexpected findings demonstrate that although the leader cell may have a pivotal role in local cell guidance, physical forces that it generates are but a small part of a global tug-of-war involving cells well back from the leading edge.