A central problem in the study of Anosov flows is to understand which closed 3-manifolds admit such a flow. We address this problem by employing Dehn–Fried surgery. Our main result shows that for every genus , there exists a finite-index subgroup  of  such that every element of  is represented by a mapping class  for which the corresponding mapping torus  carries a transitive Anosov flow.This talk is based on joint work with François Béguin, Christian Bonatti, and Bin Yu.