There is a deep connection between curvature prescription problems in conformal geometry and sharp Sobolev inequalities. The most famous example arises in the Yamabe Problem, where one observes a connection between constructing conformal metrics with constant scalar curvature and the sharp $L^2$-Sobolev inequality. A similar idea arises in Perelman's work on the Ricci flow, where one observes a connection between prescribing Perelman's weighted scalar curvature and the sharp logarithmic Sobolev inequality. In this talk, I will describe an approach to conformal geometry on smooth metric measure spaces which unifies these two ideas. In particular, I will describe how smooth metric measure spaces give a natural geometric interpretation to the family of sharp Gagliardo--Nirenberg inequalities discovered by Del Pino and Dolbeault which interpolates between the Yamabe Problem and a similar problem involving Perelman's nu-entropy.