I will present and discuss some results and problems about flows of metrics on Riemannian manifolds correlated to Ricci flow: - The "renormalization group" flow, truncated at the second order term. The Ricci flow is its trucation at the first order (joint work with L. Cremaschi). - The "Ricci-Bourguignon" flow, which is a perturbation of the Ricci flow equation by an extra term proportional to the product of the scalar curvature with the metric tensor (joint work with G. Catino, L. Cremaschi, Z. Djadli, L. Mazzieri). - A "noname" flow that I and Nicola Gigli introduced using the theory of optimal transport of mass, which is "tangent" to the Ricci flow at the initial time and which can be defined also for nonsmooth metric spaces.