In this paper, complex Lie groups acting transitively and effectively on complex manifolds with solvable (nilpotent) fundamental groups are studied. It is shown that if is nilpotent, then locally , where is semisimple and is nilpotent. In the case when is solvable, the Levi decomposition of the group is direct if and only if the stationary subgroup contains a maximal unipotent subgroup of the semisimple part. The question of the existence of transitive semisimple groups on is considered.