In this interesting paper, locally free actions of the group GA of orientation preserving affine transformations of \({\mathbb{R}}\) on closed three-manifolds are considered. If M is a closed 3-dimensional manifold with \(H^ 1(M, {\mathbb{R}})=0\), then any locally free \(C^ 2\)-action of GA on M preserves a \(C^ 0\)-volume form (Theorem D). Any locally free \(C^ r\)-action (r\(\geq 2)\) of GA on any closed 3-manifold which preserves a \(C^ 0\)-volume form is \(C^{r-1}\)-conjugate to a ''homogeneous action'', i.e., an action of the form GA\(\times G/\Gamma \to G/\Gamma\), where G is a Lie group containing GA as a subgroup, \(\Gamma\) is a discrete uniform subgroup of G and the action is induced by left translations on G (Theorem B). These are the main results of the article. However, it contains several other interesting results. Among the others: (1) homogeneous actions of GA on three manifolds are classified; (2) it is proved that any \(C^ 0\)- volume form preserved by a locally free \(C^ r\)-action of a non- unimodular Lie group G on a closed manifold M is of the class \(C^{r-2}\) if only dim M\(=\dim G+1\); (3) structural stability of some actions of fundamental groups of closed oriented surfaces of genus \(g\geq 2\) on \(S^ 1\) is established. In the proofs, several deep results of the geometric theory of dynamical systems, ergodic theory and the theory of foliations are exploited.