Ricci curvature is a fundamental notion in Riemanian geometry, which is still lacking a similar fundamental role in noncommutative geometry. In this talk I will discuss
a notion of lower bound for Ricci curvature in the noncommutative setting, and explain how this notion, motivated by the work of Villani and Lott on displacement convexity, can
be justified. Moreover, as in the commutative case it will be shown that strictly positive Ricci curvature implies very strong concentration inequalities. Last but not least we will
discuss new examples from quantum information theory.

This is joint wok with Haojian Li and Nick LaRacuente.