Recent advances about analysis on spaces with Ricci curvature bounded from below

I will review some recent result about analysis on Lott-Sturm-Villani spaces with Ricci curvature bounded from below and geometric consequences that can be derived. A crucial role in the discussion will be played by the definition of spaces with Riemannian Ricci curvature bounded from below (introduced in collaboration with Ambrosio and Savare'): this is a class of spaces strictly smaller than the one of CD(K,\infty) spaces which is still stable w.r.t. measured-Gromov-Hausdorff convergence and rules out Finsler geometries.