The classical Lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants. When applied to continuous systems, one obtains the Hopf formula relating the singularities of a flow to the Euler characteristic of the underlying manifold. The main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold. There are serious obstruction to finding such a formula for flows. The first is that there are flows on any closed manifold of dimension > 4 of zero Euler characteristic with no periodic orbits whatsoever [27]. These flows, however, are unstable by Pugh's C a closing lemma [19]. So it is natural to restrict attention to flows that satisfy Axiom A and the No Cycles property, since Smale has shown that these flows are O-stable [23]. Secondly there are gradient flows on any closed manifold that are Axiom A-No Cycles and have no closed orbits [24]. So it is natural to study only nonsingular flows. The possibility that a Lefschetz type formula for such flows existed was suggested to us by R. Bowen. Even nonsingular Morse-Smale flows, however, are difficult to deal with since Asimov's geometric index is not an invariant of the nonsingular homotopy class [2]. We must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold. We define in Sect. 2 the class of homology proper flows. Such a flow has the property that closed orbits of large period lie in large homology classes. In Sect. 7 we show that a homology proper Axiom A-No Cycles flow does satisfy a Lefschetz-type formula that relates invariants of the closed orbits to a manifold invariant. This formula is closely related to Franks' study of Smale flows on S 3 in which the linking numbers of closed orbits are used to compute the Alexander polynomial of a certain link. Indeed the manifold invariant used in Sect. 7 is a close analogue of an Alexander polynomial that we call the Alexander