In this talk we shall be focused on a quasilinear reaction diffusion system with cross diffusion, which was first proposed by Shigesada, Kawasaki and Teramoto in 1979 for investigating the spatial segregation of two competing species under inter- and intra-species population pressures. I shall talk about some recent research progress on the existence and stability of nontrivial steady states and travelling waves for the S-K-T competition model with cross diffusion, which may correspond to some new pattern formation and wave phenomena induced by cross diffusion.