A fundamental problem in Riemannian geometry is the search for the best metric on a given manifold. This, as well as problems arising in general relativity, has led to extensive research on Einstein metrics and their various generalizations. One such generalization is the relatively recent notion of quasi-Einstein metrics. In this talk, we will discuss the existence problem and structural results for left-invariant quasi-Einstein metrics on solvable Lie groups. Based on joint work with Nazia Valiyakath.