We will present our joint work with Dima Shlyakhtenko and Alice Guionnet on free transport with an emphasis on the relevant classes of (tracial) non-commutative functions.

We will start by recalling briefly classical optimal transportation and state our result on transport of free Gibbs states associated to regular h-convex potentials.

Then we will explain the notion of convexity we need, we call h-convex functions. This is a strengthening of tracial convexity involving Haagerup tensor products. We will explain the few examples we know and how we use h-convexity.

Finally, we will introduce the classes of tracial $C^k$ functions we use for free transport and the associated differential calculus.