Ricci solitons and blow-down analysis

In this expository paper, we discuss some recent developments on Ricci solitons and ancient solutions in the Ricci flow. We begin with variously typical examples of Ricci solitons and ancient solutions, then focus on some important topics in Ricci flow, such as the rigidity problem of steady gradient Ricci solitons, classification of blow-up solutions, and asymptotic behavior analysis. We also present and discuss some open problems in this area.