In this talk, we introduce and discuss a conformally invariant notion of an $m$-quasi-Einstein manifold, which interpolates between Einstein warped products with $m$-dimensional fibers and, when $m=\infty$, gradient Ricci solitons. We use this framework to prove a precompactness result for quasi- Einstein manifolds, which in particular says that compact, noncollapsed $m$- quasi-Einstein metrics can and do converge to gradient Ricci solitons. We accomplish this by introducing a functional which variationally characterizes quasi-Einstein metrics and interpolates between the Yamabe functional and Perelman's $F$ and $W$-functionals, and by introducing the corresponding "energy," which interpolates between the Yamabe constant and Perelman's $ u$-entropy. This yields a similar $\kappa$-noncollapsing result as Perelman, which allows us to then use the ideas of Anderson, Cao-Sesum, Zhang, and Weber in proving similar precompactness results.