In　this　paper　we　study　the　relation　between　Gorenstein　derived　and　pure　derived　categories　under　some　conditions.　Then,　we　also　study　the　relation　between　Gorenstein　singularity　and　pure　singularity　categories.　Furthermore,　we　describe　the　bounded　Gorenstein　derived　category　by　pure　projective　modules,　under　a　fairly　strong　assumption　on　the　ring.　In　the　last　part　of　the　paper,　we　study　the　pure　derived　category　over　a　hereditary　ring.