In　this　paper,　we　introduce　the　concept　of　a　g-Besselian　frame　in　a　Hilbert　space　and　discuss　the　relations　between　a　g-Besselian　frame　and　a　Besselian　frame.　We　also　give　some　characterizations　of　g-Besselian　frames.　In　the　end　of　this　paper,　we　discuss　the　stability　of　g-Besselian　frames.　Our　results　show　that　the　relations　and　the　characterizations　between　a　g-Besselian　frame　and　a　Besselian　frame　are　different　from　the　corresponding　results　of　g-frames　and　frames.　©　2010　Institute　of　Mathematics,　Academy　of　Mathematics　and　Systems　Science,　Chinese　Academy　of　Sciences,　Chinese　Mathematical　Society　and　Springer-Verlag　Berlin　Heidelberg.