  The purpose of this article is to study the existence and uniqueness of
quasi-Einstein structures on $3$-dimensional homogeneous Riemannian manifolds.
To this end, we use the eight model geometries for 3-dimensional manifolds
identified by Thurston. First, we present here a complete description of
quasi-Einstein metrics on $3$-dimensional homogeneous manifolds with isometry
group of dimension $4.$ In addition, we shall show the absence of such gradient
structure on $Sol^3,$ which has $3$-dimensional isometry group. Moreover, we
prove that Berger's spheres carry a non-trivial quasi-Einstein structure with
non gradient associated vector field, this shows that a theorem due to Perelman
can not be extend to quasi-Einstein metrics. Finally, we prove that a
$3$-dimensional homogeneous manifold carrying a gradient quasi-Einstein
structure is either Einstein or $\mathbb{H}^2_{\kappa} \times \mathbb{R}.$
