I will present a new set of functional inequalities involving the following four parameters associated with a given multidimensional distribution: the relative entropy, the relative Fisher information, the 2-Wasserstein distance, and the Stein discrepancy (which naturally appears in the well-known Stein's method for normal approximations). Our results improve the classical log-Sobolev and Talagrand's transport inequalities, and provide new key tools in order to deal with high-dimensional quantitative central limit theorems on a Gaussian space. Some open problems will be discussed in the last part of the talk. Joint works with M. Ledoux (Toulouse), I. Nourdin (Luxembourg) and Y. Swan (Liège).