Abstract: In this talk I shall talk about our recent work on the existence and stability/instability of several types of steady states for the SKT competition models with cross-diffusion. By applying special decomposition argument and detailed asymptotic analysis, we proved the existence and the asymptotic structure of several global branches of nontrivial steady states for two types of limiting systems and the original SKT model when one of the cross-diffusion parameter is large enough. The stability/ instability of some steady states with special local bifurcating structure or with blowing up structure are also proved by applying some special decomposition argument and the detailed spectral analysis.