This talk discusses a new non-asymptotic, "local" approach to quantitative propagation of chaos for a wide class of mean field diffusive dynamics. For a system of \$n\$ interacting particles, the relative entropy between the marginal law of \$k\$ particles and its limiting product measure is shown to be \$O((k/n)^2)\$ at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The main assumption is that the limiting measure obeys a certain functional inequality, which is shown to encompass many potentially irregular but not too singular finite-range interactions, as well as some infinite-range interactions. This unifies the previously disparate cases of Lipschitz versus bounded measurable interactions, improving the best prior bounds of \$O(k/n)\$ which were deduced from "global" estimates involving all \$n\$ particles. At the center of the new approach is a differential inequality, derived from a form of the BBGKY hierarchy, which bounds the \$k\$-particle entropy in terms of the \$(k+1)\$-particle entropy.