Classification of 4-dim gradient Ricci solitons is important to the
study of 4-dim Ricci flow with surgeries. My talk will be based on our classification of anti-self-dual gradient shrinking Ricci solitons and our results on anti-self-dual steady Ricci solitons. This is highly related to the analyticity of Ricci solitons. I will also discuss something on anti-self-dual Ricci flows.