Homogeneous structures of $3$-dimensional Lie groups

We give a classification of homogeneous Riemannian structures on (non locally symmetric) $3$-dimensional Lie groups equipped with left invariant Riemannian metrics. This work together with classifications due to previous works yields a complete classification of all the homogeneous Riemannian structures on homogeneous Riemannian $3$-spaces. Two applications of the classification to contact Riemannian geometry and CR geometry are also given.