The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent Lie groups.

35 pages. Submitted to the Proceedings of the Sixth Workshop on Lie Theory and Geometry, Cordoba (Argentina), November 13-17, 2007