The　elastic　properties　and　electronic　structure　of　the　solid　solution　of　CeO2-M　(M　=　Ti,　Zr,　Hf,　V,　Nb,　Ta)　are　studied　using　first　principle　methods　based　on　density　functional　theory　(DFT).　The　investigation　shows　that　the　elastic　constants　(B,　G,　E,　C-11　and　C-12)　of　the　solid　solution　structure　of　the　CeO2-M　system　have　increased,　resulting　in　improved　mechanical　properties,　particularly　for　CeO2-Zr.　The　elastic　constant　C-44　in　CeO2-Ti　and　CeO2-V　shows　a　marked　reduction.　The　elastic　modulus　of　CeO2-Nb　significantly　increased,　and　its　anisotropy　is　not　significantly　different　from　CeO2.　By　calculating　the　energy　band　and　density　of　states,　it　is　shown　that　CeO2　is　an　insulator,　and　there　are　two　band　gaps.　For　transition　metal　solid　solutions,　the　lattice　constant　of　the　CeO2-M　system　decreases,　and　the　second　band　gap　disappears　because　the　Ce　atomic　contribution　reduces,　and　there　is　overlap　of　the　4f　and　5d　orbital　energies.　The　valence　and　conduction　bands　of　the　CeO2-M　system　(Ti,　Zr,　Hf)　were　found　to　shift　slightly,　with　the　band　gaps　reducing　to　2.03　eV,　2.477　eV,　and　2.44　eV　respectively.　The　band　structure　of　the　CeO2-M　system　(V,　Nb,　Ta)　as　a　whole　moves　down,　and　the　second　band　gap　disappears.　For　bands　near　the　Fermi　level,　the　band　gaps　are　1.37　eV,　2.51　eV,　and　2.73　eV　respectively.　(C)　2014　Elsevier　B.V.　All　rights　reserved.