The　electronic　structure　and　formation　energies　of　defect　in　Cr-doped　ZnO　are　investigated　by　the　first-principles　calculations.　The　dopant　Cr　atom　and　native　defects　(O　vacancy,　Zn　vacancy,　and　Zn　interstitial)　at　different　charge　states　are　considered.　The　calculated　results　indicate　that　the　substitutive　Cr　in　the　oxidation　state　of　+3　and　the　neutral　Zn　vacancy　in　the　ZnO:　Cr　sample　is　the　most　favorable　defect　complex　to　maintain　a　high　stability　of　ferromagnetic　order.　Therefore,　a　p-type　ZnO:　Cr　prepared　under　O-rich　condition　could　have　a　higher　Curie　temperature.　The　p-d　hybridization　and　the　itinerant　d-characteristic　n-type　carriers　are　suggested　to　mediate　the　long　range　ferromagnetic　exchange　interaction　between　Cr　atoms.　(C)　2012　American　Institute　of　Physics.　[http://dx.doi.org/10.1063/1.4727951]