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In this paper, a complete classification of Smale flows on S3 is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstractdoi:10.1090/s0002-9947-1987-0896023-7 fatcat:zams4dbkbjasplxunfq2cug4xi