In　this　paper,　we　introduce　the　pre-frame　operator　Q　for　the　g-frame　in　a　complex　Hilbert　space,　which　will　play　a　key　role　in　studying　g-frames　and　g-Riesz　bases　etc.　Using　the　pre-frame　operator　Q,　we　give　some　necessary　and　sufficient　conditions　for　a　g-Bessel　sequence,　a　g-frame,　and　a　g-Riesz　basis　in　a　complex　Hilbert　space,　which　have　properties　similar　to　those　of　the　Bessel　sequence,　frame,　and　Riesz　basis　respectively.　We　also　obtain　the　relation　between　a　g-frame　and　a　g-Riesz　basis,　and　the　relation　of　bounds　between　a　g-frame　and　a　g-Riesz　basis.　Lastly,　we　consider　the　stability　of　a　g-frame　or　a　g-Riesz　basis　for　a　Hilbert　space　under　perturbation.　©　2008　Institute　of　Mathematics,　Academy　of　Mathematics　and　Systems　Science,　Chinese　Academy　of　Sciences,　Chinese　Mathematical　Society　and　Springer-Verlag　GmbH.