Graphene　and　metal　oxide　nanocomposites　have　been　demonstrated　as　promising　electrode　materials　for　high-performance　lithium-ion　batteries　(LIBs)　and　sodium-ion　batteries　(SIBs).　In　this　work,　ultrafine　CoMn2O4　nanoparticles　uniformly　anchored　on　reduced　graphene　oxide　(rGO)　sheets　have　been　synthesized　through　a　facile　and　effective　two-step　strategy.　Owing　to　the　rational　combination　of　merits　from　both　ternary　CoMn2O4　and　graphene　sheets,　the　as-synthesized　rGO/CoMn2O4　nanocomposite　exhibits　remarkable　Li-battery　performance　with　high　reversible　capacity,　good　cycling　stability　and　excellent　rate　performance.　Remarkably,　the　rGO/CoMn2O4　nanocomposite　displays　high　reversible　capacities　of　1102.1　and　811.1　mA　h　g(-1)　at　the　current　densities　of　200　and　500　mA　g(-1)　after　60　cycles,　respectively.　The　discharge　capacities　of　the　rGO/CoMn2O4　nanocomposite　are　as　high　as　851.1,　835.3,　795.2,　755.9,　694.0,　and　563.6　mA　h　g(-1)　at　the　current　densities　of　100,　200,　500,　1000,　2000　and　5000　mA　g(-1),　respectively.　These　electrochemical　results　suggest　the　rGO/CoMn2O4　nanocomposite　could　be　a　promising　anode　material　for　high-performance　LIBs.　Besides,　the　rGO/CoMn2O4　nanocomposite　also　exhibits　comparably　promising　electrochemical　performance　as　an　anode　material　for　SIBs.　Our　study　also　highlights　the　importance　of　rational　synthesis　of　graphene-based　nanocomposite　materials　for　high-performance　LIBs　and　SIBs.　(C)　2017　Elsevier　B.V.　All　rights　reserved.