Quantum fluctuations and geometry: From graph counting to ricci flow

We discuss in rather general terms quantum field theories dealing with spaces of maps between Riemannian manifolds. In particular we explore the well-known connection between the renormalization group flow for nonlinear sigma models and the Ricci flow. This is an expanded version of the invited talk that one of us (M.C.), has presented at The Jubilee 40th Symposium on Mathematical Physics “Geometry and Quanta” (Toruń, June 25–28, 2008).