The cone of moving curves on algebraic varieties We give a new description of the closed cone of moving curves of a smooth Fano three or fourfold by finitely many linear equations. In particular the cone is polyhedral. The proof in the threefold case relies on a famous result of Bucksom Demailly Paun and Peternell which says that the cone of moving curves is dual to the cone of pseudoeffective divisors. Additionally the proof in the fourfold case uses a result of Kawamata which describes the exceptional locus and the flip of a small contraction on a smooth fourfold. This proof provides an inductive way to compute the cone of moving curves and gives a description of the Mori cone of the variety obtained by the the flip of a small contraction.