A torsion-free abelian group \(G\) is called fully transitive (transitive) if for any elements \(0\neq a\), \(b\in G\) with height sequences \(h^ G(a)\leq h^ G(b)\) \((h^ G(a)=h^ G(b))\) there exists an endomorphism (automorphism) \(\alpha\in\hbox{End}(G)\) \((\alpha\in \hbox{Aut} G)\) such that \(\alpha a=b\). An abelian group \(G\) is called \(E\)-transitive (resp. strongly homogeneous) if the endomorphism ring \(\hbox{End} (G)\) (resp. the automorphism group \(\hbox{Aut}G\)) acts transitively on the pure rank 1 subgroups of \(G\). \(E\)-transitive (resp. strongly homogeneous) groups are homogeneous fully transitive (transitive). Fully transitive (transitive) abelian torsion-free groups are investigated in this paper. The main results are as follows. Theorem 1. The following properties of a torsion-free abelian group \(G\) are equivalent: 1) \(G\) is a fully transitive group and \(G=\hbox{Soc} G\); 2) \(G=\oplus_{p\in\Pi}G_ p\) for some set \(\Pi\) of primes of \(\mathbb{Z}\) and \(\Pi(G_ p)\cap\Pi(G_ q)=\emptyset\) for \(p\neq q\) \((p,q\in\Pi)\) and \(G_ p\) is a homogeneous fully transitive group or \(G_ p\) is a non- homogeneous fully transitive group such that the quasi-endomorphism ring \(\mathcal C(G_ p)\) is a skew field \((\Pi(G_ q)=\{p\mid pG_ q\neq G_ q\})\). A component \(G_ p\) of a fully transitive group \(G\) is called finite type subgroup if \(G_ p\) is a non-homogeneous group or \(G_ p\) is homogeneous, and \(\hbox{rank}_ C G_ p<\infty\) where \(C=\hbox{End}_ RG_ p\), \(R=\hbox{End}(G)\). Theorem 2. Let \(G\) be a torsion-free abelian fully transitive group with finiteftype component \(G_ p\). Then \(G=G_ p\oplus H\) for some pure fully invariant subgroup \(H\) such that \(\Pi(G_ p)\cap\Pi(H)=\emptyset\) and \(H\) being unique.