K-frames,　which　are　a　generalization　of　frames,　allow　us　in　a　stable　way,　to　reconstruct　elements　from　the　range　of　a　linear　and　bounded　operator　in　a　Hilbert　space.　In　this　paper,　we　first　make　a　discussion　on　relations　between　K-frames　and　frames.　We　obtain　that　a　tight　K-frame　turns　to　be　a　frame　if　and　only　if　the　linear　and　bounded　operator　K　is　surjective,　and　then　give　a　necessary　and　sufficient　condition　for　a　K-frame　with　closed　range　operator　K.　We　also　study　the　constructions　of　K-frames　under　some　conditions.　Finally　we　study　the　stabilities　of　K-frames　under　small　perturbations.　©,　2014,　Chinese　Academy　of　Sciences.　All　right　reserved.