For the flows in nozzle with a contact discontinuity in between may be subsonic-subsonic or subsonic-supersonic or supersonic-supersonic. In this talk, we will talk about the cases for the subsonic-supersonic and supersonic-supersonic contact discontinuity in a finitely long nozzle and establish the stabilities for both of them by applying the characteristic methods as well as the theories for the second order elliptic equation. This talk is based on joint works with Prof. F. M. Huang, Prof. D. H. Wang and Prof. W. Xiang.