It is ``well-known and elementary'' that Anosov flows have trivial centralizer (Ghys); we (Bakker, Fisher and Hasselblatt) prove this for transitive expansive flows. Our main result gives a residual set of non-Anosov $C^\infty$ Axiom A flows with no cycles such that diffeomorphisms commuting with them are time-$t$ maps of the flow. This requires a study of the Lie group of commuting diffeomorphisms using the flow dynamics on the invariant "foliations." The novelty is that we study flows and under weaker dynamical assumptions.