In this talk, I will present some recent results on the dynamic and steady states for the degenerate Keller-Segel (KS) system with general fast and slow diffusion exponents. The KS system is a basic mathematical model describing the biological phenomenon chemotaxis. We will discuss some behaviors such as the hyper-contractivity and ultra-contracivity, existence of the global weak entropy solution, finite time extinction, finite-time blow-up,shape asymptotic behaviors and related to the critical exponents in steady solutions. This is a joint work with Shen Bian of Tsinghua University.