We say an $n$-dimensional Riemannian manifold is an $m$-Quasi Einstein metric if it is the base of an $(n+m)$-dimensional warped product Einstein manifold. We view the $m$-Quasi Einstein equation as a generalization of the Einstein equation (since an Einstein manifold is the base of a trivial product Einstein manifold). The $m$-Quasi Einstein equation is also closely related to the gradient Ricci soliton equation. In this talk I will give an overview of some earlier results about the classification of $m$-quasi Einstein metrics and prove a new classification of $m$-Quasi Einstein metrics with harmonic curvature. This is joint work with Peter Petersen and Chenxu He.