We　introduce　the　concepts　of　K-g-frames　and　K-g-Riesz　bases　in　complex　Hilbert　spaces　and　study　the　essential　distinctions　between　a　K-g-frame　and　a　g-frame.　We　give　a　characterization　of　a　K-g-frames　and　an　equivalent　characterization　of　a　K-g-Riesz　basis　in　complex　Hilbert　spaces.　Lastly,　by　using　the　analytical　methods　and　techniques　in　frame　theory,　we　obtain　some　sufficient　conditions　for　the　stability　of　perturbation　of　K-g-frames　and　K-g-Riesz　bases.　©,　2014,　Chinese　Academy　of　Sciences.　All　right　reserved.