THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs.   We will discuss conformally flat complete Yamabe flow and show that in some cases we can give the precise description of singularity profiles close to the extinction time of the solution. We will also talk about a construction of new compact ancient solutions to the Yamabe flow. This is a joint work with Daskalopoulos, King and Manuel del Pino.