according to perelman's work on ricci flow with surgeries in dimension 3, we know that it is important to understand at least qualitative behaviors of singularity formation in order to perform surgeries. the situation in dimension 4 is much more complicated as some new types of singularity models may arise and the classification of the singularity models is far from complete. we expect that the singularity models should be solitons, self-similar solutions to the ricci flow, and we expect that most of singularity models are shrinking gradient solitons with possible singularities by the recent work of richard bamler. steady gradient ricci solitons may also arise as singularity models and they are related to shrinking solitons with quadratic curvature growth. in a recent joint work with r. bamler, b. chow, y. deng, and y. zhang, we managed to classify tangent flows at infinity which can be viewed as a blow-down of 4 dimensional steady gradient ricci soliton singularity models. when the tangent flow at infinity is 3-cylindrical, we can give very good qualitative characterization of such steady solitons. we will also mention the somewhat parallel work with y. zhang on the existence of asymptotic shrinkers on steady solitons with nonnegative ricci curvature.