Abstract: The existence of a transport map from the standard Gaussian leads to succinct representations for, potentially complicated, measures. Inspired by results from optimal transport, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space. Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz when the target measure satisfies an appropriate convexity assumption. This facilitates the proof of several new functional inequalities. In other settings, where a globally Lipschitz transport map cannot exist, we derive Sobolev estimates which are intimately connected to several famous open problems in convex geometry. Joint work with Yair Shenfeld