In　this　paper,　we　introduce　the　more　general　g-frame　which　is　called　a　K-g-frame　by　combining　a　g-frame　with　a　bounded　linear　operator　K　in　a　Hilbert　space.　We　give　several　equivalent　characterizations　for　K-g-frames　and　discuss　the　stability　of　perturbation　for　K-g-frames.　We　also　investigate　the　relationship　between　a　K-g-frame　and　the　range　of　the　bounded　linear　operator　K.　In　the　end,　we　give　two　sufficient　conditions　for　the　remainder　of　a　K-g-frame　after　an　erasure　to　still　be　a　K-g-frame.　It　turns　out　that　although　K-g-frames　share　some　properties　similar　to　g-frames,　a　large　part　of　K-g-frames　behaves　completely　different　from　g-frames.　Copyright　©　2015　Rocky　Mountain　Mathematics　Consortium.