Abstract: In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p} Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences and a priori $L^p$ scalar curvature bounds for $p<1$ This is joint work with Man-Chun Lee and Aaron Naber.