Let $F={F^v:S^1->S^1,vin; V$ and $g={G^v:S^1->S^1,vin; V$ be disjoint flows defined on the unit circle $S^1$, that is such flows that each their element either is the identity mapping or has no fixed point ((V, +) is a 2-divisible nontrivial abelian group). The aim of this paper is to give a necessary and sufficient codition for topological conjugacy of disjoint flows i.e., the existence of a homeomorphism $Gamma:S^1->S^1$ satisfying $$Gammacirc F^v=G^vcircGamma,; vin; V$$ Moreover, under some further restrictions, we determine all such homeomorphisms.