In Perelman's well-known paper, he claimed the unqueness of the 3D \verb=\=kappa-noncollpased steady GRS (without giving any proof or ideas) and conjectured that 3D noncompact \verb=\=kappa-solution with positive sectional curvature must be the Ricci flow generated the Bryant soliton. Perelman's claim and conjecture has been proved by Simon Brendle in 2012 and 2018, respectively. The classification of 3D \verb=\=kappa-noncollpased steady GRS plays an important role in his proof of the conjecture. Brendle's work is based on the observation that 3D \verb=\=kappa-noncollpased steady GRS must be asymptotically cylindercial. In higher dimensions, the asymptotical geometry of steady GRS is much more complicated. In this talk, we will talk about some recent progress on the asymptotical geometry of 4D steady GRS. This is a joint work with Prof. Bennett Chow.