  A Lie group $G$ endowed with a left invariant Riemannian metric $g$ is called
Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds
is an important area of investigation. In this paper, we study different
aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups.
We show that this class of biharmonic maps can be used at the first level to
build examples but, as we will see through this paper, its study will lead to
some interesting mathematical problems in the theory of Riemannian Lie groups.
